// https://judge.eluminatis-of-lu.com/contest/676ffd92569fb90008aac7da/1157
use crate::algo_lib::collections::treap::payload::{OrdPayload, Payload};
use crate::algo_lib::collections::treap::Tree;
use crate::algo_lib::collections::vec_ext::inc_dec::IncDec;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use crate::algo_lib::graph::Graph;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::recursive_function::{Callable2, RecursiveFunction2};
use crate::algo_lib::misc::test_type::TaskType;
use crate::algo_lib::misc::test_type::TestType;
type PreCalc = ();
fn solve(input: &mut Input, out: &mut Output, _test_case: usize, _data: &mut PreCalc) {
    let n = input.read_size();
    let a = input.read_size_vec(n).dec();
    let edges = input.read_size_pair_vec(n - 1).dec();
    struct Node(usize);
    impl Payload for Node {}
    impl OrdPayload for Node {
        type Key = usize;
        fn key(&self) -> &Self::Key {
            &self.0
        }
        fn union(a: Self, _b: Self) -> Self {
            a
        }
    }
    let mut ans = vec![0; n];
    let graph = Graph::from_biedges(n, &edges);
    let mut dfs = RecursiveFunction2::new(|
        f,
        vert: usize,
        prev: usize,
    | -> (Tree<Node>, usize) {
        let mut treap = Tree::new();
        treap.insert(Node(a[vert]));
        let mut size = 1;
        for e in &graph[vert] {
            if e.to() == prev {
                continue;
            }
            let (call, call_sz) = f.call(e.to(), vert);
            treap = Tree::union(treap, call);
            size += call_sz;
        }
        ans[vert] = size - treap.range(..&size).size();
        (treap, size)
    });
    dfs.call(0, n);
    let q = input.read_size();
    for _ in 0..q {
        let x = input.read_size() - 1;
        out.print_line(ans[x]);
    }
}
pub static TEST_TYPE: TestType = TestType::MultiNumber;
pub static TASK_TYPE: TaskType = TaskType::Classic;
pub(crate) fn run(mut input: Input, mut output: Output) -> bool {
    let mut pre_calc = ();
    match TEST_TYPE {
        TestType::Single => solve(&mut input, &mut output, 1, &mut pre_calc),
        TestType::MultiNumber => {
            let t = input.read();
            for i in 1..=t {
                solve(&mut input, &mut output, i, &mut pre_calc);
            }
        }
        TestType::MultiEof => {
            let mut i = 1;
            while input.peek().is_some() {
                solve(&mut input, &mut output, i, &mut pre_calc);
                i += 1;
            }
        }
    }
    output.flush();
    match TASK_TYPE {
        TaskType::Classic => input.is_empty(),
        TaskType::Interactive => true,
    }
}


fn main() {
    let mut sin = std::io::stdin();
    let input = crate::algo_lib::io::input::Input::new(&mut sin);
    let mut stdout = std::io::stdout();
    let output = crate::algo_lib::io::output::Output::new(&mut stdout);
    run(input, output);
}
pub mod algo_lib {
pub mod collections {
pub mod dsu {
use crate::algo_lib::collections::slice_ext::bounds::Bounds;
use crate::algo_lib::collections::slice_ext::indices::Indices;
use std::cell::Cell;
#[derive(Clone)]
pub struct DSU {
    id: Vec<Cell<i32>>,
    count: usize,
}
impl DSU {
    pub fn new(n: usize) -> Self {
        Self {
            id: vec![Cell::new(- 1); n],
            count: n,
        }
    }
    pub fn size(&self, i: usize) -> usize {
        (-self.id[self.find(i)].get()) as usize
    }
    #[allow(clippy::len_without_is_empty)]
    pub fn len(&self) -> usize {
        self.id.len()
    }
    pub fn iter(&self) -> impl Iterator<Item = usize> + '_ {
        self.id
            .iter()
            .enumerate()
            .filter_map(|(i, id)| if id.get() < 0 { Some(i) } else { None })
    }
    pub fn set_count(&self) -> usize {
        self.count
    }
    pub fn union(&mut self, mut a: usize, mut b: usize) -> bool {
        a = self.find(a);
        b = self.find(b);
        if a == b {
            false
        } else {
            self.id[a].replace(self.id[a].get() + self.id[b].get());
            self.id[b].replace(a as i32);
            self.count -= 1;
            true
        }
    }
    pub fn find(&self, i: usize) -> usize {
        if self.id[i].get() >= 0 {
            let res = self.find(self.id[i].get() as usize);
            self.id[i].replace(res as i32);
            res
        } else {
            i
        }
    }
    pub fn clear(&mut self) {
        self.count = self.id.len();
        self.id.fill(Cell::new(-1));
    }
    pub fn parts(&self) -> Vec<Vec<usize>> {
        let roots: Vec<_> = self.iter().collect();
        let mut res = vec![Vec::new(); roots.len()];
        for i in self.id.indices() {
            res[roots.as_slice().bin_search(&self.find(i)).unwrap()].push(i);
        }
        res
    }
}
}
pub mod slice_ext {
pub mod bounds {
pub trait Bounds<T: PartialOrd> {
    fn lower_bound(&self, el: &T) -> usize;
    fn upper_bound(&self, el: &T) -> usize;
    fn bin_search(&self, el: &T) -> Option<usize>;
    fn more(&self, el: &T) -> usize;
    fn more_or_eq(&self, el: &T) -> usize;
    fn less(&self, el: &T) -> usize {
        self.lower_bound(el)
    }
    fn less_or_eq(&self, el: &T) -> usize {
        self.upper_bound(el)
    }
}
impl<T: PartialOrd> Bounds<T> for [T] {
    fn lower_bound(&self, el: &T) -> usize {
        let mut left = 0;
        let mut right = self.len();
        while left < right {
            let mid = left + ((right - left) >> 1);
            if &self[mid] < el {
                left = mid + 1;
            } else {
                right = mid;
            }
        }
        left
    }
    fn upper_bound(&self, el: &T) -> usize {
        let mut left = 0;
        let mut right = self.len();
        while left < right {
            let mid = left + ((right - left) >> 1);
            if &self[mid] <= el {
                left = mid + 1;
            } else {
                right = mid;
            }
        }
        left
    }
    fn bin_search(&self, el: &T) -> Option<usize> {
        let at = self.lower_bound(el);
        if at == self.len() || &self[at] != el { None } else { Some(at) }
    }
    fn more(&self, el: &T) -> usize {
        self.len() - self.upper_bound(el)
    }
    fn more_or_eq(&self, el: &T) -> usize {
        self.len() - self.lower_bound(el)
    }
}
}
pub mod indices {
use std::ops::Range;
pub trait Indices {
    fn indices(&self) -> Range<usize>;
}
impl<T> Indices for [T] {
    fn indices(&self) -> Range<usize> {
        0..self.len()
    }
}
}
}
pub mod treap {





use crate::algo_lib::collections::treap::payload::{OrdPayload, Payload, Pushable};
use crate::algo_lib::misc::direction::Direction;
use crate::algo_lib::misc::extensions::replace_with::ReplaceWith;
use crate::algo_lib::misc::random::{gen, gen_range};
use std::collections::Bound;
use std::marker::{PhantomData, PhantomPinned};
use std::mem::{forget, replace, swap, take, MaybeUninit};
use std::ops::{Deref, DerefMut, RangeBounds};
use std::ptr::NonNull;
pub struct Content<P> {
    payload: P,
    parent: NodeLink<Node<P>>,
    left: NodeLink<Node<P>>,
    right: NodeLink<Node<P>>,
}
impl<P: Payload> Content<P> {
    fn push_down(&mut self) {
        if P::NEED_PUSH_DOWN && self.payload.need_push_down() {
            if self.left.size != 0 {
                self.left.deref_mut().payload_mut().push_delta(&self.payload);
            }
            if self.right.size != 0 {
                self.right.deref_mut().payload_mut().push_delta(&self.payload);
            }
            self.payload.reset_delta();
        }
    }
    fn update(&mut self) {
        if P::NEED_UPDATE {
            self.payload.update(self.left.payload(), self.right.payload());
        }
    }
}
pub struct Node<P> {
    priority: u64,
    size: u32,
    reversed: bool,
    content: Option<Content<P>>,
    _phantom_pinned: PhantomPinned,
}
impl<P> Node<P> {
    const NULL_NODE: Self = Self {
        priority: 0,
        size: 0,
        reversed: false,
        content: None,
        _phantom_pinned: PhantomPinned,
    };
    fn new(payload: P) -> NodeLink<Node<P>> {
        NodeLink::new(Node {
            priority: gen(),
            size: 1,
            reversed: false,
            content: Some(Content {
                payload,
                parent: NodeLink::default(),
                left: NodeLink::default(),
                right: NodeLink::default(),
            }),
            _phantom_pinned: PhantomPinned,
        })
    }
    fn payload(&self) -> Option<&P> {
        self.content.as_ref().map(|c| &c.payload)
    }
    fn payload_mut(&mut self) -> &mut P {
        unsafe { &mut self.content.as_mut().unwrap_unchecked().payload }
    }
    fn reverse(&mut self) {
        if let Some(content) = &mut self.content {
            self.reversed = !self.reversed;
            swap(&mut content.left, &mut content.right);
        }
    }
}
impl<P: Payload> Node<P> {
    fn update(&mut self) {
        self.size = 1 + self.left.size + self.right.size;
        self.deref_mut().update();
    }
    fn push_down(&mut self) {
        if self.reversed {
            self.left.reverse();
            self.right.reverse();
            self.reversed = false;
        }
        self.deref_mut().push_down();
    }
    fn detach(&mut self) {
        if self.size != 0 {
            self.parent = NodeLink::default();
        }
    }
    fn detach_left(&mut self) -> NodeLink<Node<P>> {
        self.push_down();
        let mut left = take(&mut self.left);
        left.detach();
        left
    }
    fn detach_right(&mut self) -> NodeLink<Node<P>> {
        self.push_down();
        let mut right = take(&mut self.right);
        right.detach();
        right
    }
    fn attach_left(&mut self, left: NodeLink<Node<P>>) {
        assert_eq!(self.left.size, 0);
        if left.size != 0 {
            self.left = left;
            self.left.deref_mut().parent = NodeLink::new_ref(self);
        }
        self.update();
    }
    fn attach_right(&mut self, right: NodeLink<Node<P>>) {
        assert_eq!(self.right.size, 0);
        if right.size != 0 {
            self.right = right;
            self.right.deref_mut().parent = NodeLink::new_ref(self);
        }
        self.update();
    }
    fn heapify(&mut self) {
        if self.left.size != 0 {
            self.left.heapify();
            if self.left.priority > self.priority {
                let left_priority = self.left.priority;
                self.left.priority = self.priority;
                self.priority = left_priority;
            }
        }
        if self.right.size != 0 {
            self.right.heapify();
            if self.right.priority > self.priority {
                let right_priority = self.right.priority;
                self.right.priority = self.priority;
                self.priority = right_priority;
            }
        }
    }
    fn gen(n: usize, f: impl FnMut(usize) -> P) -> NodeLink<Node<P>> {
        let mut res = Self::build(f, 0, n).0;
        res.heapify();
        res
    }
    fn build<F: FnMut(usize) -> P>(
        mut f: F,
        from: usize,
        to: usize,
    ) -> (NodeLink<Node<P>>, F) {
        if from == to {
            (NodeLink::default(), f)
        } else {
            let mid = gen_range(from..to);
            let mut node = Node::new(f(mid));
            let (left, f) = Self::build(f, from, mid);
            let (right, f) = Self::build(f, mid + 1, to);
            node.attach_left(left);
            node.attach_right(right);
            (node, f)
        }
    }
    fn push<D>(&mut self, delta: D)
    where
        P: Pushable<D>,
    {
        self.payload_mut().push(delta);
    }
    fn refs(&mut self, res: &mut Vec<NodeId<P>>) {
        if self.size != 0 {
            self.left.refs(res);
            res.push(NodeId(NodeLink::new_ref(self)));
            self.right.refs(res);
        }
    }
    fn first(&mut self) -> &Node<P> {
        if self.size == 0 {
            &Self::NULL_NODE
        } else {
            self.push_down();
            if self.left.size != 0 { self.left.first() } else { self }
        }
    }
    fn last(&mut self) -> &Node<P> {
        if self.size == 0 {
            &Self::NULL_NODE
        } else {
            self.push_down();
            if self.right.size != 0 { self.right.last() } else { self }
        }
    }
    fn binary_search<F: FnMut(&P, Option<&P>, Option<&P>) -> Option<Direction>>(
        &mut self,
        mut f: F,
    ) {
        if self.size != 0 {
            self.push_down();
            let direction = f(&self.payload, self.left.payload(), self.right.payload());
            match direction {
                Some(Direction::Left) => self.left.binary_search(f),
                Some(Direction::Right) => self.right.binary_search(f),
                None => {}
            }
        }
    }
    fn size(&self) -> usize {
        self.size as usize
    }
    fn binary_search_size<F: FnMut(usize, usize) -> Option<Direction>>(
        &mut self,
        mut f: F,
    ) {
        if self.size != 0 {
            self.push_down();
            let direction = f(self.left.size(), self.right.size());
            match direction {
                Some(Direction::Left) => self.left.binary_search_size(f),
                Some(Direction::Right) => self.right.binary_search_size(f),
                None => {}
            }
        }
    }
}
impl<P> Deref for Node<P> {
    type Target = Content<P>;
    fn deref(&self) -> &Self::Target {
        unsafe { self.content.as_ref().unwrap_unchecked() }
    }
}
impl<P> DerefMut for Node<P> {
    fn deref_mut(&mut self) -> &mut Self::Target {
        unsafe { self.content.as_mut().unwrap_unchecked() }
    }
}
pub struct NodeId<P>(NodeLink<Node<P>>);
pub struct NodeLink<P> {
    link: NonNull<P>,
}
impl<P> NodeLink<Node<P>> {
    fn new(node: Node<P>) -> Self {
        let mut pinned = Box::pin(node);
        let res = NodeLink {
            link: unsafe { NonNull::from(pinned.as_mut().get_unchecked_mut()) },
        };
        forget(pinned);
        res
    }
    fn new_ref(node: &Node<P>) -> Self {
        NodeLink {
            link: NonNull::from(node),
        }
    }
    fn into_payload(mut self) -> P {
        assert_eq!(self.left.size, 0);
        assert_eq!(self.right.size, 0);
        assert_eq!(self.parent.size, 0);
        self.content.take().unwrap().payload
    }
}
impl<P: Payload> NodeLink<Node<P>> {
    fn merge(mut left: Self, mut right: Self) -> Self {
        match (left.size, right.size) {
            (0, _) => right,
            (_, 0) => left,
            _ => {
                if left.priority > right.priority {
                    let left_right = left.detach_right();
                    left.attach_right(Self::merge(left_right, right));
                    left
                } else {
                    let right_left = right.detach_left();
                    right.attach_left(Self::merge(left, right_left));
                    right
                }
            }
        }
    }
    fn split_by<F: FnMut(&P, Option<&P>, Option<&P>) -> Direction>(
        mut self,
        mut f: F,
    ) -> (Self, Self) {
        if self.size != 0 {
            let direction = f(&self.payload, self.left.payload(), self.right.payload());
            match direction {
                Direction::Left => {
                    let (left, right) = self.detach_left().split_by(f);
                    self.attach_left(right);
                    (left, self)
                }
                Direction::Right => {
                    let (left, right) = self.detach_right().split_by(f);
                    self.attach_right(left);
                    (self, right)
                }
            }
        } else {
            (NodeLink::default(), NodeLink::default())
        }
    }
    fn push_from_up(&self, directions: &mut Vec<Direction>) -> NodeLink<Node<P>> {
        if self.parent.size != 0 {
            if self.parent.left == *self {
                directions.push(Direction::Left);
            } else if self.parent.right == *self {
                directions.push(Direction::Right);
            } else {
                unreachable!();
            }
            self.parent.push_from_up(directions)
        } else {
            NodeLink { link: self.link }
        }
    }
    fn raise(self, link: &NodeLink<Node<P>>) -> (Self, Self, Self) {
        assert!(link.content.is_some());
        let mut directions = Vec::new();
        let expected_parent = link.push_from_up(&mut directions);
        assert!(expected_parent == self);
        self.split_by_dir(directions)
    }
    fn split_by_dir(mut self, mut directions: Vec<Direction>) -> (Self, Self, Self) {
        if let Some(dir) = directions.pop() {
            match dir {
                Direction::Left => {
                    let (left, mid, right) = self.detach_left().split_by_dir(directions);
                    self.attach_left(right);
                    (left, mid, self)
                }
                Direction::Right => {
                    let (left, mid, right) = self
                        .detach_right()
                        .split_by_dir(directions);
                    self.attach_right(left);
                    (self, mid, right)
                }
            }
        } else {
            let left = self.detach_left();
            let right = self.detach_right();
            self.update();
            (left, self, right)
        }
    }
    fn split_by_size<F: FnMut(usize, usize) -> Direction>(
        mut self,
        mut f: F,
    ) -> (Self, Self) {
        if self.size != 0 {
            let direction = f(self.left.size(), self.right.size());
            match direction {
                Direction::Left => {
                    let (left, right) = self.detach_left().split_by_size(f);
                    self.attach_left(right);
                    (left, self)
                }
                Direction::Right => {
                    let (left, right) = self.detach_right().split_by_size(f);
                    self.attach_right(left);
                    (self, right)
                }
            }
        } else {
            (Self::default(), Self::default())
        }
    }
    fn split_at(self, mut at: usize) -> (Self, Self) {
        self.split_by_size(|left_size, _| {
            if at <= left_size {
                Direction::Left
            } else {
                at -= left_size + 1;
                Direction::Right
            }
        })
    }
}
impl<P: OrdPayload> NodeLink<Node<P>> {
    fn split(self, key: &P::Key) -> (Self, Self) {
        self.split_by(|payload, _left, _right| {
            if payload.key() < key { Direction::Right } else { Direction::Left }
        })
    }
    fn split_inclusive(self, key: &P::Key) -> (Self, Self) {
        self.split_by(|payload, _left, _right| {
            if payload.key() <= key { Direction::Right } else { Direction::Left }
        })
    }
    fn union(mut a: Self, mut b: Self) -> Self {
        match (a.size, b.size) {
            (0, _) => b,
            (_, 0) => a,
            _ => {
                if a.priority < b.priority {
                    swap(&mut a, &mut b);
                }
                let (b_left, b_right) = b.split(a.payload.key());
                let (same, b_right) = b_right.split_inclusive(a.payload.key());
                let (left, right) = if same.size != 0 {
                    let left = a.detach_left();
                    let right = a.detach_right();
                    a = Node::new(P::union(a.into_payload(), same.into_payload()));
                    (left, right)
                } else {
                    (a.detach_left(), a.detach_right())
                };
                let left = Self::union(left, b_left);
                let right = Self::union(right, b_right);
                a.attach_left(left);
                a.attach_right(right);
                a
            }
        }
    }
}
impl<P> Clone for NodeLink<P> {
    fn clone(&self) -> Self {
        NodeLink { link: self.link }
    }
}
impl<P> PartialEq for NodeLink<P> {
    fn eq(&self, other: &Self) -> bool {
        self.link == other.link
    }
}
impl<P> Eq for NodeLink<P> {}
impl<P> Deref for NodeLink<P> {
    type Target = P;
    fn deref(&self) -> &Self::Target {
        unsafe { self.link.as_ref() }
    }
}
impl<P> DerefMut for NodeLink<P> {
    fn deref_mut(&mut self) -> &mut Self::Target {
        unsafe { self.link.as_mut() }
    }
}
impl<P> Default for NodeLink<Node<P>> {
    fn default() -> Self {
        NodeLink {
            link: unsafe {
                NonNull::new_unchecked(
                    &Node::NULL_NODE as *const Node<P> as *mut Node<P>,
                )
            },
        }
    }
}
pub enum Tree<P> {
    Whole { root: NodeLink<Node<P>> },
    Split { left: NodeLink<Node<P>>, mid: Box<Tree<P>>, right: NodeLink<Node<P>> },
}
impl<P: Payload> Default for Tree<P> {
    fn default() -> Self {
        Self::new()
    }
}
impl<P: Payload> Tree<P> {
    pub fn new() -> Self {
        Tree::Whole {
            root: NodeLink::default(),
        }
    }
    pub fn gen(n: usize, f: impl FnMut(usize) -> P) -> Self {
        Self::gen_impl(n, f)
    }
    fn single(p: P) -> Self {
        Tree::Whole { root: Node::new(p) }
    }
    fn gen_impl(n: usize, f: impl FnMut(usize) -> P) -> Self {
        Tree::Whole {
            root: Node::gen(n, f),
        }
    }
    fn into_node(mut self) -> NodeLink<Node<P>> {
        self.rebuild();
        match self {
            Tree::Whole { root } => root,
            _ => unreachable!(),
        }
    }
    fn as_node(&mut self) -> &mut NodeLink<Node<P>> {
        match self {
            Tree::Whole { root } => root,
            _ => unreachable!(),
        }
    }
    fn rebuild(&mut self) -> &mut NodeLink<Node<P>> {
        self.replace_with(|self_| {
            if let Tree::Split { left, mid, right } = self_ {
                Tree::Whole {
                    root: NodeLink::merge(left, NodeLink::merge(mid.into_node(), right)),
                }
            } else {
                self_
            }
        });
        self.as_node()
    }
    fn do_split(
        mut self,
        f: impl FnOnce(NodeLink<Node<P>>) -> (NodeLink<Node<P>>, NodeLink<Node<P>>),
    ) -> (Self, Self) {
        let mut right = MaybeUninit::uninit();
        self.replace_with(|self_| {
            let root = self_.into_node();
            let (left, right_) = f(root);
            right.write(Tree::Whole { root: right_ });
            Tree::Whole { root: left }
        });
        (self, unsafe { right.assume_init() })
    }
    fn do_split_three(
        &mut self,
        f: impl FnOnce(
            NodeLink<Node<P>>,
        ) -> (NodeLink<Node<P>>, NodeLink<Node<P>>, NodeLink<Node<P>>),
    ) -> &mut Self {
        self.replace_with(|self_| {
            let root = self_.into_node();
            let (left, mid, right) = f(root);
            Self::Split {
                left,
                mid: Box::new(Self::Whole { root: mid }),
                right,
            }
        });
        self.as_mid()
    }
    fn as_mid(&mut self) -> &mut Self {
        match self {
            Tree::Split { mid, .. } => mid,
            _ => unreachable!(),
        }
    }
    pub fn push_front(&mut self, other: Self) -> &mut Self {
        self.replace_with(|root| Tree::Whole {
            root: NodeLink::merge(other.into_node(), root.into_node()),
        });
        self
    }
    pub fn push_back(&mut self, other: Self) -> &mut Self {
        self.replace_with(|root| Tree::Whole {
            root: NodeLink::merge(root.into_node(), other.into_node()),
        });
        self
    }
    pub fn detach(&mut self) -> Self {
        match self {
            Tree::Whole { root } => Tree::Whole { root: take(root) },
            Tree::Split { mid, .. } => replace(mid, Tree::new()),
        }
    }
    pub fn binary_search(
        &mut self,
        f: impl FnMut(&P, Option<&P>, Option<&P>) -> Option<Direction>,
    ) {
        self.rebuild().binary_search(f);
    }
    pub fn push<D>(&mut self, delta: D)
    where
        P: Pushable<D>,
    {
        self.rebuild().push(delta);
    }
    pub fn merge(left: Self, right: Self) -> Self {
        match left {
            Tree::Whole { root: left_root } => {
                Tree::Split {
                    left: left_root,
                    mid: Box::new(right),
                    right: NodeLink::default(),
                }
            }
            Tree::Split { left, mid, right: left_right } => {
                Tree::Split {
                    left,
                    mid,
                    right: NodeLink::merge(left_right, right.into_node()),
                }
            }
        }
    }
    pub fn merge_three(left: Self, mid: Self, right: Self) -> Self {
        Self::Split {
            left: left.into_node(),
            mid: Box::new(mid),
            right: right.into_node(),
        }
    }
    pub fn iter(&mut self) -> Iter<'_, P> {
        Iter::new(self.rebuild())
    }
    pub fn first(&mut self) -> Option<&P> {
        self.rebuild().first().payload()
    }
    pub fn last(&mut self) -> Option<&P> {
        self.rebuild().last().payload()
    }
    pub fn payload(&mut self) -> Option<&P> {
        self.rebuild().payload()
    }
    pub fn add_back(&mut self, payload: P) -> NodeId<P> {
        let mut res = MaybeUninit::uninit();
        self.replace_with(|root| {
            let mut new_node = Self::single(payload);
            res.write(NodeId(new_node.as_node().clone()));
            Self::merge(root, new_node)
        });
        unsafe { res.assume_init() }
    }
    pub fn add_front(&mut self, payload: P) -> NodeId<P> {
        let mut res = MaybeUninit::uninit();
        self.replace_with(|root| {
            let mut new_node = Self::single(payload);
            res.write(NodeId(new_node.as_node().clone()));
            Self::merge(new_node, root)
        });
        unsafe { res.assume_init() }
    }
    pub fn refs(&mut self) -> Vec<NodeId<P>> {
        let mut res = Vec::new();
        self.rebuild().refs(&mut res);
        res
    }
    pub fn is_empty(&self) -> bool {
        match self {
            Tree::Whole { root } => root.size == 0,
            Tree::Split { left, mid, right } => {
                left.size == 0 && right.size == 0 && mid.is_empty()
            }
        }
    }
    pub fn raise(&mut self, node_ref: &NodeId<P>) -> &mut Self {
        self.do_split_three(|node| node.raise(&node_ref.0))
    }
    pub fn into_payload(self) -> P {
        self.into_node().into_payload()
    }
    pub fn index_ref(&mut self, node_ref: &NodeId<P>) -> usize {
        self.raise(node_ref);
        match self {
            Tree::Whole { .. } => unreachable!(),
            Tree::Split { left, .. } => left.size(),
        }
    }
    pub fn size(&self) -> usize {
        match self {
            Tree::Whole { root } => root.size(),
            Tree::Split { left, mid, right } => left.size() + mid.size() + right.size(),
        }
    }
    pub fn split_at(self, at: usize) -> (Self, Self) {
        self.do_split(|root| root.split_at(at))
    }
    pub fn binary_search_size(
        &mut self,
        f: impl FnMut(usize, usize) -> Option<Direction>,
    ) {
        self.rebuild().binary_search_size(f);
    }
    pub fn range_index(&mut self, bounds: impl RangeBounds<usize>) -> &mut Self {
        self.do_split_three(|root| {
            let size = root.size();
            let start = match bounds.start_bound() {
                Bound::Included(&s) => s,
                Bound::Excluded(&s) => s + 1,
                Bound::Unbounded => 0,
            };
            let end = match bounds.end_bound() {
                Bound::Included(&e) => e + 1,
                Bound::Excluded(&e) => e,
                Bound::Unbounded => size,
            };
            assert!(start <= end);
            let (left, mid_right) = root.split_at(start);
            let (mid, right) = mid_right.split_at(end.max(start) - start);
            (left, mid, right)
        })
    }
    pub fn get_at(&mut self, at: usize) -> &mut Self {
        self.range_index(at..=at)
    }
    pub fn reverse(&mut self) {
        self.rebuild().reverse();
    }
}
pub struct Iter<'a, P> {
    stack: Vec<*mut NodeLink<Node<P>>>,
    _marker: PhantomData<&'a P>,
}
impl<'a, P: Payload> Iter<'a, P> {
    fn new(root: &'a mut NodeLink<Node<P>>) -> Self {
        let mut res = Self {
            stack: Vec::new(),
            _marker: PhantomData,
        };
        res.add_left(root);
        res
    }
    fn add_left(&mut self, mut node: &mut NodeLink<Node<P>>) {
        while node.size != 0 {
            node.push_down();
            self.stack.push(node);
            node = &mut node.left;
        }
    }
}
impl<'a, P: Payload> Iterator for Iter<'a, P> {
    type Item = &'a P;
    fn next(&mut self) -> Option<Self::Item> {
        let node = unsafe { &mut *self.stack.pop()? };
        self.add_left(&mut node.right);
        Some(&node.payload)
    }
}
impl<P: OrdPayload> Tree<P> {
    pub fn range<'s: 'r, 'r>(
        &'s mut self,
        bounds: impl RangeBounds<&'r P::Key>,
    ) -> &mut Self {
        self.do_split_three(|root| {
            let (left, mid_right) = match bounds.start_bound() {
                Bound::Included(key) => root.split(key),
                Bound::Excluded(key) => root.split_inclusive(key),
                Bound::Unbounded => (NodeLink::default(), root),
            };
            let (mid, right) = match bounds.end_bound() {
                Bound::Included(key) => mid_right.split_inclusive(key),
                Bound::Excluded(key) => mid_right.split(key),
                Bound::Unbounded => (mid_right, NodeLink::default()),
            };
            (left, mid, right)
        })
    }
    pub fn insert(&mut self, p: P) -> Option<P> {
        let mid = self.range(&p.key()..=&p.key());
        let mut res = None;
        mid.replace_with(|mid| {
            if mid.size() != 0 {
                res = Some(mid.into_node().into_payload());
            }
            Tree::single(p)
        });
        res
    }
    pub fn remove(&mut self, key: &P::Key) -> Option<P> {
        let mid = self.range(key..=key);
        let mut res = None;
        mid.replace_with(|mid| {
            if mid.size() != 0 {
                res = Some(mid.into_node().into_payload());
            }
            Self::new()
        });
        res
    }
    pub fn split(self, key: &P::Key) -> (Self, Self) {
        self.do_split(|root| root.split(key))
    }
    pub fn split_inclusive(self, key: &P::Key) -> (Self, Self) {
        self.do_split(|root| root.split_inclusive(key))
    }
    pub fn get(&mut self, key: &P::Key) -> Option<&P> {
        self.range(key..=key).payload()
    }
    pub fn floor(&mut self, key: &P::Key) -> Option<&P> {
        self.range(..=key).last()
    }
    pub fn ceil(&mut self, key: &P::Key) -> Option<&P> {
        self.range(key..).first()
    }
    pub fn prev(&mut self, key: &P::Key) -> Option<&P> {
        self.range(..key).last()
    }
    pub fn next(&mut self, key: &P::Key) -> Option<&P> {
        self.range((Bound::Excluded(key), Bound::Unbounded)).first()
    }
    pub fn union(a: Self, b: Self) -> Self {
        let a = a.into_node();
        let b = b.into_node();
        Self::Whole {
            root: NodeLink::union(a, b),
        }
    }
    pub fn index(&mut self, key: &P::Key) -> Option<usize> {
        match self.range(key..=key) {
            Tree::Split { left, mid, .. } => mid.payload().map(|_| left.size()),
            _ => unreachable!(),
        }
    }
}
pub mod payload {
#[allow(unused_variables)]
pub trait Payload: Sized {
    const NEED_UPDATE: bool = false;
    const NEED_PUSH_DOWN: bool = false;
    fn reset_delta(&mut self) {
        unimplemented!()
    }
    fn update(&mut self, left: Option<&Self>, right: Option<&Self>) {
        unimplemented!()
    }
    fn push_delta(&mut self, delta: &Self) {
        unimplemented!()
    }
    fn need_push_down(&self) -> bool {
        unimplemented!()
    }
}
#[allow(unused_variables)]
pub trait OrdPayload: Payload {
    type Key: Ord;
    fn key(&self) -> &Self::Key;
    fn union(a: Self, b: Self) -> Self {
        unimplemented!()
    }
}
pub trait Pushable<Delta>: Payload {
    fn push(&mut self, delta: Delta);
}
impl<P: Payload> Pushable<&P> for P {
    #[inline]
    fn push(&mut self, delta: &P) {
        self.push_delta(delta);
    }
}
impl<P: Payload> Pushable<P> for P {
    #[inline]
    fn push(&mut self, delta: P) {
        *self = delta;
    }
}
}
}
pub mod vec_ext {
pub mod default {
pub fn default_vec<T: Default>(len: usize) -> Vec<T> {
    let mut v = Vec::with_capacity(len);
    for _ in 0..len {
        v.push(T::default());
    }
    v
}
}
pub mod inc_dec {
use crate::algo_lib::numbers::num_traits::algebra::{AdditionMonoidWithSub, One};
pub trait IncDec {
    #[must_use]
    fn inc(self) -> Self;
    #[must_use]
    fn dec(self) -> Self;
}
impl<T: AdditionMonoidWithSub + One> IncDec for T {
    fn inc(self) -> Self {
        self + T::one()
    }
    fn dec(self) -> Self {
        self - T::one()
    }
}
impl<T: AdditionMonoidWithSub + One> IncDec for Vec<T> {
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|i| *i += T::one());
        self
    }
    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|i| *i -= T::one());
        self
    }
}
impl<T: AdditionMonoidWithSub + One> IncDec for Vec<Vec<T>> {
    fn inc(mut self) -> Self {
        self.iter_mut().for_each(|v| v.iter_mut().for_each(|i| *i += T::one()));
        self
    }
    fn dec(mut self) -> Self {
        self.iter_mut().for_each(|v| v.iter_mut().for_each(|i| *i -= T::one()));
        self
    }
}
impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One> IncDec
for Vec<(T, U)> {
    fn inc(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j)| {
                *i += T::one();
                *j += U::one();
            });
        self
    }
    fn dec(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j)| {
                *i -= T::one();
                *j -= U::one();
            });
        self
    }
}
impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V> IncDec
for Vec<(T, U, V)> {
    fn inc(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j, _)| {
                *i += T::one();
                *j += U::one();
            });
        self
    }
    fn dec(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j, _)| {
                *i -= T::one();
                *j -= U::one();
            });
        self
    }
}
impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V, W> IncDec
for Vec<(T, U, V, W)> {
    fn inc(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j, ..)| {
                *i += T::one();
                *j += U::one();
            });
        self
    }
    fn dec(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j, ..)| {
                *i -= T::one();
                *j -= U::one();
            });
        self
    }
}
impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V, W, X> IncDec
for Vec<(T, U, V, W, X)> {
    fn inc(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j, ..)| {
                *i += T::one();
                *j += U::one();
            });
        self
    }
    fn dec(mut self) -> Self {
        self.iter_mut()
            .for_each(|(i, j, ..)| {
                *i -= T::one();
                *j -= U::one();
            });
        self
    }
}
impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One> IncDec for (T, U) {
    fn inc(mut self) -> Self {
        self.0 += T::one();
        self.1 += U::one();
        self
    }
    fn dec(mut self) -> Self {
        self.0 -= T::one();
        self.1 -= U::one();
        self
    }
}
}
}
}
pub mod graph {
use crate::algo_lib::collections::dsu::DSU;
use crate::algo_lib::graph::edges::bi_edge::BiEdge;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::edges::edge_trait::{BidirectionalEdgeTrait, EdgeTrait};
use std::ops::{Index, IndexMut};



















#[derive(Clone)]
pub struct Graph<E: EdgeTrait> {
    edges: Vec<Vec<E>>,
    edge_count: usize,
}
impl<E: EdgeTrait> Graph<E> {
    pub fn new(vertex_count: usize) -> Self {
        Self {
            edges: vec![Vec::new(); vertex_count],
            edge_count: 0,
        }
    }
    pub fn add_edge(&mut self, (from, mut edge): (usize, E)) -> usize {
        let to = edge.to();
        assert!(to < self.vertex_count());
        let direct_id = self.edges[from].len();
        edge.set_id(self.edge_count);
        self.edges[from].push(edge);
        if E::REVERSABLE {
            let rev_id = self.edges[to].len();
            self.edges[from][direct_id].set_reverse_id(rev_id);
            let mut rev_edge = self.edges[from][direct_id].reverse_edge(from);
            rev_edge.set_id(self.edge_count);
            rev_edge.set_reverse_id(direct_id);
            self.edges[to].push(rev_edge);
        }
        self.edge_count += 1;
        direct_id
    }
    pub fn add_vertices(&mut self, cnt: usize) {
        self.edges.resize(self.edges.len() + cnt, Vec::new());
    }
    pub fn clear(&mut self) {
        self.edge_count = 0;
        for ve in self.edges.iter_mut() {
            ve.clear();
        }
    }
    pub fn vertex_count(&self) -> usize {
        self.edges.len()
    }
    pub fn edge_count(&self) -> usize {
        self.edge_count
    }
    pub fn degrees(&self) -> Vec<usize> {
        self.edges.iter().map(|v| v.len()).collect()
    }
}
impl<E: BidirectionalEdgeTrait> Graph<E> {
    pub fn is_tree(&self) -> bool {
        if self.edge_count + 1 != self.vertex_count() {
            false
        } else {
            self.is_connected()
        }
    }
    pub fn is_forest(&self) -> bool {
        let mut dsu = DSU::new(self.vertex_count());
        for i in 0..self.vertex_count() {
            for e in self[i].iter() {
                if i <= e.to() && !dsu.union(i, e.to()) {
                    return false;
                }
            }
        }
        true
    }
    pub fn is_connected(&self) -> bool {
        let mut dsu = DSU::new(self.vertex_count());
        for i in 0..self.vertex_count() {
            for e in self[i].iter() {
                dsu.union(i, e.to());
            }
        }
        dsu.set_count() == 1
    }
}
impl<E: EdgeTrait> Index<usize> for Graph<E> {
    type Output = [E];
    fn index(&self, index: usize) -> &Self::Output {
        &self.edges[index]
    }
}
impl<E: EdgeTrait> IndexMut<usize> for Graph<E> {
    fn index_mut(&mut self, index: usize) -> &mut Self::Output {
        &mut self.edges[index]
    }
}
impl Graph<Edge<()>> {
    pub fn from_edges(n: usize, edges: &[(usize, usize)]) -> Self {
        let mut graph = Self::new(n);
        for &(from, to) in edges {
            graph.add_edge(Edge::new(from, to));
        }
        graph
    }
}
impl<P: Clone> Graph<Edge<P>> {
    pub fn from_edges_with_payload(n: usize, edges: &[(usize, usize, P)]) -> Self {
        let mut graph = Self::new(n);
        for (from, to, p) in edges.iter() {
            graph.add_edge(Edge::with_payload(*from, *to, p.clone()));
        }
        graph
    }
}
impl Graph<BiEdge<()>> {
    pub fn from_biedges(n: usize, edges: &[(usize, usize)]) -> Self {
        let mut graph = Self::new(n);
        for &(from, to) in edges {
            graph.add_edge(BiEdge::new(from, to));
        }
        graph
    }
}
impl<P: Clone> Graph<BiEdge<P>> {
    pub fn from_biedges_with_payload(n: usize, edges: &[(usize, usize, P)]) -> Self {
        let mut graph = Self::new(n);
        for (from, to, p) in edges.iter() {
            graph.add_edge(BiEdge::with_payload(*from, *to, p.clone()));
        }
        graph
    }
}
pub mod edges {
pub mod bi_edge {
use crate::algo_lib::graph::edges::bi_edge_trait::BiEdgeTrait;
use crate::algo_lib::graph::edges::edge_id::{EdgeId, NoId, WithId};
use crate::algo_lib::graph::edges::edge_trait::{BidirectionalEdgeTrait, EdgeTrait};
#[derive(Clone)]
pub struct BiEdgeRaw<Id: EdgeId, P> {
    to: u32,
    id: Id,
    payload: P,
}
impl<Id: EdgeId> BiEdgeRaw<Id, ()> {
    pub fn new(from: usize, to: usize) -> (usize, Self) {
        (
            from,
            Self {
                to: to as u32,
                id: Id::new(),
                payload: (),
            },
        )
    }
}
impl<Id: EdgeId, P> BiEdgeRaw<Id, P> {
    pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
        (from, Self::with_payload_impl(to, payload))
    }
    fn with_payload_impl(to: usize, payload: P) -> BiEdgeRaw<Id, P> {
        Self {
            to: to as u32,
            id: Id::new(),
            payload,
        }
    }
}
impl<Id: EdgeId, P: Clone> BidirectionalEdgeTrait for BiEdgeRaw<Id, P> {}
impl<Id: EdgeId, P: Clone> EdgeTrait for BiEdgeRaw<Id, P> {
    type Payload = P;
    const REVERSABLE: bool = true;
    fn to(&self) -> usize {
        self.to as usize
    }
    fn id(&self) -> usize {
        self.id.id()
    }
    fn set_id(&mut self, id: usize) {
        self.id.set_id(id);
    }
    fn reverse_id(&self) -> usize {
        panic!("no reverse id")
    }
    fn set_reverse_id(&mut self, _: usize) {}
    fn reverse_edge(&self, from: usize) -> Self {
        Self::with_payload_impl(from, self.payload.clone())
    }
    fn payload(&self) -> &P {
        &self.payload
    }
}
impl<Id: EdgeId, P: Clone> BiEdgeTrait for BiEdgeRaw<Id, P> {}
pub type BiEdge<P> = BiEdgeRaw<NoId, P>;
pub type BiEdgeWithId<P> = BiEdgeRaw<WithId, P>;
}
pub mod bi_edge_trait {
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
pub trait BiEdgeTrait: EdgeTrait {}
}
pub mod edge {
use crate::algo_lib::graph::edges::edge_id::{EdgeId, NoId, WithId};
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
#[derive(Clone)]
pub struct EdgeRaw<Id: EdgeId, P> {
    to: u32,
    id: Id,
    payload: P,
}
impl<Id: EdgeId> EdgeRaw<Id, ()> {
    pub fn new(from: usize, to: usize) -> (usize, Self) {
        (
            from,
            Self {
                to: to as u32,
                id: Id::new(),
                payload: (),
            },
        )
    }
}
impl<Id: EdgeId, P> EdgeRaw<Id, P> {
    pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
        (from, Self::with_payload_impl(to, payload))
    }
    fn with_payload_impl(to: usize, payload: P) -> Self {
        Self {
            to: to as u32,
            id: Id::new(),
            payload,
        }
    }
}
impl<Id: EdgeId, P: Clone> EdgeTrait for EdgeRaw<Id, P> {
    type Payload = P;
    const REVERSABLE: bool = false;
    fn to(&self) -> usize {
        self.to as usize
    }
    fn id(&self) -> usize {
        self.id.id()
    }
    fn set_id(&mut self, id: usize) {
        self.id.set_id(id);
    }
    fn reverse_id(&self) -> usize {
        panic!("no reverse")
    }
    fn set_reverse_id(&mut self, _: usize) {
        panic!("no reverse")
    }
    fn reverse_edge(&self, _: usize) -> Self {
        panic!("no reverse")
    }
    fn payload(&self) -> &P {
        &self.payload
    }
}
pub type Edge<P> = EdgeRaw<NoId, P>;
pub type EdgeWithId<P> = EdgeRaw<WithId, P>;
}
pub mod edge_id {
pub trait EdgeId: Clone {
    fn new() -> Self;
    fn id(&self) -> usize;
    fn set_id(&mut self, id: usize);
}
#[derive(Clone)]
pub struct WithId {
    id: u32,
}
impl EdgeId for WithId {
    fn new() -> Self {
        Self { id: 0 }
    }
    fn id(&self) -> usize {
        self.id as usize
    }
    fn set_id(&mut self, id: usize) {
        self.id = id as u32;
    }
}
#[derive(Clone)]
pub struct NoId {}
impl EdgeId for NoId {
    fn new() -> Self {
        Self {}
    }
    fn id(&self) -> usize {
        panic!("Id called on no id")
    }
    fn set_id(&mut self, _: usize) {}
}
}
pub mod edge_trait {
pub trait EdgeTrait: Clone {
    type Payload;
    const REVERSABLE: bool;
    fn to(&self) -> usize;
    fn id(&self) -> usize;
    fn set_id(&mut self, id: usize);
    fn reverse_id(&self) -> usize;
    fn set_reverse_id(&mut self, reverse_id: usize);
    #[must_use]
    fn reverse_edge(&self, from: usize) -> Self;
    fn payload(&self) -> &Self::Payload;
}
pub trait BidirectionalEdgeTrait: EdgeTrait {}
}
}
}
pub mod io {
pub mod input {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::io::Read;
use std::mem::MaybeUninit;
pub struct Input<'s> {
    input: &'s mut (dyn Read + Send),
    buf: Vec<u8>,
    at: usize,
    buf_read: usize,
    eol: bool,
}
macro_rules! read_impl {
    ($t:ty, $read_name:ident, $read_vec_name:ident) => {
        pub fn $read_name (& mut self) -> $t { self.read() } pub fn $read_vec_name (& mut
        self, len : usize) -> Vec <$t > { self.read_vec(len) }
    };
    ($t:ty, $read_name:ident, $read_vec_name:ident, $read_pair_vec_name:ident) => {
        read_impl!($t, $read_name, $read_vec_name); pub fn $read_pair_vec_name (& mut
        self, len : usize) -> Vec < ($t, $t) > { self.read_vec(len) }
    };
}
impl<'s> Input<'s> {
    const DEFAULT_BUF_SIZE: usize = 4096;
    pub fn new(input: &'s mut (dyn Read + Send)) -> Self {
        Self {
            input,
            buf: default_vec(Self::DEFAULT_BUF_SIZE),
            at: 0,
            buf_read: 0,
            eol: true,
        }
    }
    pub fn new_with_size(input: &'s mut (dyn Read + Send), buf_size: usize) -> Self {
        Self {
            input,
            buf: default_vec(buf_size),
            at: 0,
            buf_read: 0,
            eol: true,
        }
    }
    pub fn get(&mut self) -> Option<u8> {
        if self.refill_buffer() {
            let res = self.buf[self.at];
            self.at += 1;
            if res == b'\r' {
                self.eol = true;
                if self.refill_buffer() && self.buf[self.at] == b'\n' {
                    self.at += 1;
                }
                return Some(b'\n');
            }
            self.eol = res == b'\n';
            Some(res)
        } else {
            None
        }
    }
    pub fn peek(&mut self) -> Option<u8> {
        if self.refill_buffer() {
            let res = self.buf[self.at];
            Some(if res == b'\r' { b'\n' } else { res })
        } else {
            None
        }
    }
    pub fn skip_whitespace(&mut self) {
        while let Some(b) = self.peek() {
            if !b.is_ascii_whitespace() {
                return;
            }
            self.get();
        }
    }
    pub fn next_token(&mut self) -> Option<Vec<u8>> {
        self.skip_whitespace();
        let mut res = Vec::new();
        while let Some(c) = self.get() {
            if c.is_ascii_whitespace() {
                break;
            }
            res.push(c);
        }
        if res.is_empty() { None } else { Some(res) }
    }
    pub fn is_exhausted(&mut self) -> bool {
        self.peek().is_none()
    }
    pub fn is_empty(&mut self) -> bool {
        self.skip_whitespace();
        self.is_exhausted()
    }
    pub fn read<T: Readable>(&mut self) -> T {
        T::read(self)
    }
    pub fn read_vec<T: Readable>(&mut self, size: usize) -> Vec<T> {
        let mut res = Vec::with_capacity(size);
        for _ in 0..size {
            res.push(self.read());
        }
        res
    }
    pub fn read_char(&mut self) -> u8 {
        self.skip_whitespace();
        self.get().unwrap()
    }
    read_impl!(u32, read_unsigned, read_unsigned_vec);
    read_impl!(u64, read_u64, read_u64_vec);
    read_impl!(usize, read_size, read_size_vec, read_size_pair_vec);
    read_impl!(i32, read_int, read_int_vec, read_int_pair_vec);
    read_impl!(i64, read_long, read_long_vec, read_long_pair_vec);
    read_impl!(i128, read_i128, read_i128_vec);
    fn refill_buffer(&mut self) -> bool {
        if self.at == self.buf_read {
            self.at = 0;
            self.buf_read = self.input.read(&mut self.buf).unwrap();
            self.buf_read != 0
        } else {
            true
        }
    }
    pub fn is_eol(&self) -> bool {
        self.eol
    }
}
pub trait Readable {
    fn read(input: &mut Input) -> Self;
}
impl Readable for u8 {
    fn read(input: &mut Input) -> Self {
        input.read_char()
    }
}
impl<T: Readable> Readable for Vec<T> {
    fn read(input: &mut Input) -> Self {
        let size = input.read();
        input.read_vec(size)
    }
}
impl<T: Readable, const SIZE: usize> Readable for [T; SIZE] {
    fn read(input: &mut Input) -> Self {
        unsafe {
            let mut res = MaybeUninit::<[T; SIZE]>::uninit();
            for i in 0..SIZE {
                let ptr: *mut T = (*res.as_mut_ptr()).as_mut_ptr();
                ptr.add(i).write(input.read::<T>());
            }
            res.assume_init()
        }
    }
}
macro_rules! read_integer {
    ($($t:ident)+) => {
        $(impl Readable for $t { fn read(input : & mut Input) -> Self { input
        .skip_whitespace(); let mut c = input.get().unwrap(); let sgn = match c { b'-' =>
        { c = input.get().unwrap(); true } b'+' => { c = input.get().unwrap(); false } _
        => false, }; let mut res = 0; loop { assert!(c.is_ascii_digit()); res *= 10; let
        d = (c - b'0') as $t; if sgn { res -= d; } else { res += d; } match input.get() {
        None => break, Some(ch) => { if ch.is_ascii_whitespace() { break; } else { c =
        ch; } } } } res } })+
    };
}
read_integer!(i8 i16 i32 i64 i128 isize u16 u32 u64 u128 usize);
macro_rules! tuple_readable {
    ($($name:ident)+) => {
        impl <$($name : Readable),+> Readable for ($($name,)+) { fn read(input : & mut
        Input) -> Self { ($($name ::read(input),)+) } }
    };
}
tuple_readable! {
    T
}
tuple_readable! {
    T U
}
tuple_readable! {
    T U V
}
tuple_readable! {
    T U V X
}
tuple_readable! {
    T U V X Y
}
tuple_readable! {
    T U V X Y Z
}
tuple_readable! {
    T U V X Y Z A
}
tuple_readable! {
    T U V X Y Z A B
}
tuple_readable! {
    T U V X Y Z A B C
}
tuple_readable! {
    T U V X Y Z A B C D
}
tuple_readable! {
    T U V X Y Z A B C D E
}
tuple_readable! {
    T U V X Y Z A B C D E F
}
impl Read for Input<'_> {
    fn read(&mut self, buf: &mut [u8]) -> std::io::Result<usize> {
        if self.at == self.buf_read {
            self.input.read(buf)
        } else {
            let mut i = 0;
            while i < buf.len() && self.at < self.buf_read {
                buf[i] = self.buf[self.at];
                i += 1;
                self.at += 1;
            }
            Ok(i)
        }
    }
}
}
pub mod output {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::cmp::Reverse;
use std::io::{stderr, Stderr, Write};
#[derive(Copy, Clone)]
pub enum BoolOutput {
    YesNo,
    YesNoCaps,
    PossibleImpossible,
    Custom(&'static str, &'static str),
}
impl BoolOutput {
    pub fn output(&self, output: &mut Output, val: bool) {
        (if val { self.yes() } else { self.no() }).write(output);
    }
    fn yes(&self) -> &str {
        match self {
            BoolOutput::YesNo => "Yes",
            BoolOutput::YesNoCaps => "YES",
            BoolOutput::PossibleImpossible => "Possible",
            BoolOutput::Custom(yes, _) => yes,
        }
    }
    fn no(&self) -> &str {
        match self {
            BoolOutput::YesNo => "No",
            BoolOutput::YesNoCaps => "NO",
            BoolOutput::PossibleImpossible => "Impossible",
            BoolOutput::Custom(_, no) => no,
        }
    }
}
pub struct Output<'s> {
    output: &'s mut dyn Write,
    buf: Vec<u8>,
    at: usize,
    auto_flush: bool,
    bool_output: BoolOutput,
    precision: Option<usize>,
    separator: u8,
}
impl<'s> Output<'s> {
    const DEFAULT_BUF_SIZE: usize = 4096;
    pub fn new(output: &'s mut dyn Write) -> Self {
        Self {
            output,
            buf: default_vec(Self::DEFAULT_BUF_SIZE),
            at: 0,
            auto_flush: false,
            bool_output: BoolOutput::YesNoCaps,
            precision: None,
            separator: b' ',
        }
    }
    pub fn new_with_auto_flush(output: &'s mut dyn Write) -> Self {
        Self {
            output,
            buf: default_vec(Self::DEFAULT_BUF_SIZE),
            at: 0,
            auto_flush: true,
            bool_output: BoolOutput::YesNoCaps,
            precision: None,
            separator: b' ',
        }
    }
    pub fn flush(&mut self) {
        if self.at != 0 {
            self.output.write_all(&self.buf[..self.at]).unwrap();
            self.output.flush().unwrap();
            self.at = 0;
        }
    }
    pub fn print<T: Writable>(&mut self, s: T) {
        s.write(self);
        self.maybe_flush();
    }
    pub fn print_line<T: Writable>(&mut self, s: T) {
        self.print(s);
        self.put(b'\n');
        self.maybe_flush();
    }
    pub fn put(&mut self, b: u8) {
        self.buf[self.at] = b;
        self.at += 1;
        if self.at == self.buf.len() {
            self.flush();
        }
    }
    pub fn maybe_flush(&mut self) {
        if self.auto_flush {
            self.flush();
        }
    }
    pub fn print_per_line<T: Writable>(&mut self, arg: &[T]) {
        self.print_per_line_iter(arg.iter());
    }
    pub fn print_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        let mut first = true;
        for e in iter {
            if first {
                first = false;
            } else {
                self.put(self.separator);
            }
            e.write(self);
        }
    }
    pub fn print_line_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        self.print_iter(iter);
        self.put(b'\n');
    }
    pub fn print_per_line_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        for e in iter {
            e.write(self);
            self.put(b'\n');
        }
    }
    pub fn set_bool_output(&mut self, bool_output: BoolOutput) {
        self.bool_output = bool_output;
    }
    pub fn set_precision(&mut self, precision: usize) {
        self.precision = Some(precision);
    }
    pub fn reset_precision(&mut self) {
        self.precision = None;
    }
    pub fn get_precision(&self) -> Option<usize> {
        self.precision
    }
    pub fn separator(&self) -> u8 {
        self.separator
    }
    pub fn set_separator(&mut self, separator: u8) {
        self.separator = separator;
    }
}
impl Write for Output<'_> {
    fn write(&mut self, buf: &[u8]) -> std::io::Result<usize> {
        let mut start = 0usize;
        let mut rem = buf.len();
        while rem > 0 {
            let len = (self.buf.len() - self.at).min(rem);
            self.buf[self.at..self.at + len].copy_from_slice(&buf[start..start + len]);
            self.at += len;
            if self.at == self.buf.len() {
                self.flush();
            }
            start += len;
            rem -= len;
        }
        self.maybe_flush();
        Ok(buf.len())
    }
    fn flush(&mut self) -> std::io::Result<()> {
        self.flush();
        Ok(())
    }
}
pub trait Writable {
    fn write(&self, output: &mut Output);
}
impl Writable for &str {
    fn write(&self, output: &mut Output) {
        output.write_all(self.as_bytes()).unwrap();
    }
}
impl Writable for String {
    fn write(&self, output: &mut Output) {
        output.write_all(self.as_bytes()).unwrap();
    }
}
impl Writable for char {
    fn write(&self, output: &mut Output) {
        output.put(*self as u8);
    }
}
impl Writable for u8 {
    fn write(&self, output: &mut Output) {
        output.put(*self);
    }
}
impl<T: Writable> Writable for [T] {
    fn write(&self, output: &mut Output) {
        output.print_iter(self.iter());
    }
}
impl<T: Writable, const N: usize> Writable for [T; N] {
    fn write(&self, output: &mut Output) {
        output.print_iter(self.iter());
    }
}
impl<T: Writable + ?Sized> Writable for &T {
    fn write(&self, output: &mut Output) {
        T::write(self, output)
    }
}
impl<T: Writable> Writable for Vec<T> {
    fn write(&self, output: &mut Output) {
        self.as_slice().write(output);
    }
}
impl Writable for () {
    fn write(&self, _output: &mut Output) {}
}
macro_rules! write_to_string {
    ($($t:ident)+) => {
        $(impl Writable for $t { fn write(& self, output : & mut Output) { self
        .to_string().write(output); } })+
    };
}
write_to_string!(u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);
macro_rules! tuple_writable {
    ($name0:ident $($name:ident : $id:tt)*) => {
        impl <$name0 : Writable, $($name : Writable,)*> Writable for ($name0, $($name,)*)
        { fn write(& self, out : & mut Output) { self.0.write(out); $(out.put(out
        .separator); self.$id .write(out);)* } }
    };
}
tuple_writable! {
    T
}
tuple_writable! {
    T U : 1
}
tuple_writable! {
    T U : 1 V : 2
}
tuple_writable! {
    T U : 1 V : 2 X : 3
}
tuple_writable! {
    T U : 1 V : 2 X : 3 Y : 4
}
tuple_writable! {
    T U : 1 V : 2 X : 3 Y : 4 Z : 5
}
tuple_writable! {
    T U : 1 V : 2 X : 3 Y : 4 Z : 5 A : 6
}
tuple_writable! {
    T U : 1 V : 2 X : 3 Y : 4 Z : 5 A : 6 B : 7
}
tuple_writable! {
    T U : 1 V : 2 X : 3 Y : 4 Z : 5 A : 6 B : 7 C : 8
}
impl<T: Writable> Writable for Option<T> {
    fn write(&self, output: &mut Output) {
        match self {
            None => (-1).write(output),
            Some(t) => t.write(output),
        }
    }
}
impl Writable for bool {
    fn write(&self, output: &mut Output) {
        let bool_output = output.bool_output;
        bool_output.output(output, *self)
    }
}
impl<T: Writable> Writable for Reverse<T> {
    fn write(&self, output: &mut Output) {
        self.0.write(output);
    }
}
static mut ERR: Option<Stderr> = None;
pub fn err() -> Output<'static> {
    unsafe {
        if ERR.is_none() {
            ERR = Some(stderr());
        }
        Output::new_with_auto_flush(ERR.as_mut().unwrap())
    }
}
}
}
pub mod misc {
pub mod direction {
#[derive(Copy, Clone)]
pub enum Direction {
    Left,
    Right,
}
}
pub mod extensions {
pub mod replace_with {
use std::{mem, ptr};
pub trait ReplaceWith<F: FnOnce(Self) -> Self>: Sized {
    fn replace_with(&mut self, f: F) {
        unsafe {
            let old = ptr::read(self);
            let new = on_unwind(move || f(old), || std::process::abort());
            ptr::write(self, new);
        }
    }
}
impl<T, F: FnOnce(Self) -> Self> ReplaceWith<F> for T {}
pub fn on_unwind<F: FnOnce() -> T, T, P: FnOnce()>(f: F, p: P) -> T {
    let x = OnDrop(mem::ManuallyDrop::new(p));
    let t = f();
    let mut x = mem::ManuallyDrop::new(x);
    unsafe { mem::ManuallyDrop::drop(&mut x.0) };
    t
}
struct OnDrop<F: FnOnce()>(mem::ManuallyDrop<F>);
impl<F: FnOnce()> Drop for OnDrop<F> {
    #[inline(always)]
    fn drop(&mut self) {
        (unsafe { ptr::read(&*self.0) })();
    }
}
}
}
pub mod random {
use crate::algo_lib::collections::slice_ext::indices::Indices;
use crate::algo_lib::numbers::num_traits::algebra::IntegerSemiRingWithSub;
use crate::algo_lib::numbers::num_traits::ord::MinMax;
use crate::algo_lib::numbers::num_traits::primitive::Primitive;
use std::cell::RefCell;
use std::ops::{RangeBounds, Rem};
use std::time::SystemTime;
const NN: usize = 312;
const MM: usize = 156;
const MATRIX_A: u64 = 0xB5026F5AA96619E9;
const UM: u64 = 0xFFFFFFFF80000000;
const LM: u64 = 0x7FFFFFFF;
const F: u64 = 6364136223846793005;
const MAG01: [u64; 2] = [0, MATRIX_A];
pub struct Random {
    mt: [u64; NN],
    index: usize,
}
impl Random {
    pub fn new() -> Self {
        Self::new_with_seed(
            (SystemTime::UNIX_EPOCH.elapsed().unwrap().as_nanos() & 0xFFFFFFFFFFFFFFFF)
                as u64,
        )
    }
    pub fn new_with_seed(seed: u64) -> Self {
        let mut res = Self { mt: [0u64; NN], index: NN };
        res.mt[0] = seed;
        for i in 1..NN {
            res.mt[i] = F
                .wrapping_mul(res.mt[i - 1] ^ (res.mt[i - 1] >> 62))
                .wrapping_add(i as u64);
        }
        res
    }
    fn gen_impl(&mut self) -> u64 {
        if self.index == NN {
            for i in 0..(NN - MM) {
                let x = (self.mt[i] & UM) | (self.mt[i + 1] & LM);
                self.mt[i] = self.mt[i + MM] ^ (x >> 1) ^ MAG01[(x & 1) as usize];
            }
            for i in (NN - MM)..(NN - 1) {
                let x = (self.mt[i] & UM) | (self.mt[i + 1] & LM);
                self.mt[i] = self.mt[i + MM - NN] ^ (x >> 1) ^ MAG01[(x & 1) as usize];
            }
            let x = (self.mt[NN - 1] & UM) | (self.mt[0] & LM);
            self.mt[NN - 1] = self.mt[MM - 1] ^ (x >> 1) ^ MAG01[(x & 1) as usize];
            self.index = 0;
        }
        let mut x = self.mt[self.index];
        self.index += 1;
        x ^= (x >> 29) & 0x5555555555555555;
        x ^= (x << 17) & 0x71D67FFFEDA60000;
        x ^= (x << 37) & 0xFFF7EEE000000000;
        x ^= x >> 43;
        x
    }
    pub fn gen<T>(&mut self) -> T
    where
        u64: Primitive<T>,
    {
        self.gen_impl().to()
    }
    pub fn gen_u128(&mut self) -> u128 {
        (self.gen_impl() as u128) << 64 | self.gen_impl() as u128
    }
    pub fn gen_i128(&mut self) -> i128 {
        self.gen_u128() as i128
    }
    pub fn gen_bool(&mut self) -> bool {
        (self.gen_impl() & 1) == 1
    }
    pub fn gen_bound<T: Rem<Output = T> + Primitive<u64>>(&mut self, n: T) -> T
    where
        u64: Primitive<T>,
    {
        (self.gen_impl() % n.to()).to()
    }
    pub fn gen_range<T: IntegerSemiRingWithSub + Primitive<u64> + MinMax>(
        &mut self,
        range: impl RangeBounds<T>,
    ) -> T
    where
        u64: Primitive<T>,
    {
        let f = match range.start_bound() {
            std::ops::Bound::Included(&s) => s,
            std::ops::Bound::Excluded(&s) => s + T::one(),
            std::ops::Bound::Unbounded => T::min_val(),
        };
        let t = match range.end_bound() {
            std::ops::Bound::Included(&e) => e,
            std::ops::Bound::Excluded(&e) => e - T::one(),
            std::ops::Bound::Unbounded => T::max_val(),
        };
        if f == T::min_val() && t == T::max_val() {
            self.gen()
        } else {
            f + self.gen_bound(t - f + T::one())
        }
    }
}
thread_local! {
    static RANDOM : RefCell < Random > = RefCell::new(Random::new());
}
pub fn gen<T>() -> T
where
    u64: Primitive<T>,
{
    RANDOM.with_borrow_mut(|r| r.gen())
}
pub fn gen_u128() -> u128 {
    RANDOM.with_borrow_mut(|r| r.gen_u128())
}
pub fn gen_i128() -> i128 {
    RANDOM.with_borrow_mut(|r| r.gen_i128())
}
pub fn gen_bool() -> bool {
    RANDOM.with_borrow_mut(|r| r.gen_bool())
}
pub fn gen_bound<T: Rem<Output = T> + Primitive<u64>>(n: T) -> T
where
    u64: Primitive<T>,
{
    RANDOM.with_borrow_mut(|r| r.gen_bound(n))
}
pub fn gen_range<T: IntegerSemiRingWithSub + Primitive<u64> + MinMax>(
    range: impl RangeBounds<T>,
) -> T
where
    u64: Primitive<T>,
{
    RANDOM.with_borrow_mut(|r| r.gen_range(range))
}
pub trait Shuffle {
    fn shuffle(&mut self);
}
impl<T> Shuffle for [T] {
    fn shuffle(&mut self) {
        for i in self.indices() {
            let at = gen_bound(i + 1);
            self.swap(i, at);
        }
    }
}
}
pub mod recursive_function {
use std::marker::PhantomData;
macro_rules! recursive_function {
    ($name:ident, $trait:ident, ($($type:ident $arg:ident,)*)) => {
        pub trait $trait <$($type,)* Output > { fn call(& mut self, $($arg : $type,)*) ->
        Output; } pub struct $name < F, $($type,)* Output > where F : FnMut(& mut dyn
        $trait <$($type,)* Output >, $($type,)*) -> Output, { f : std::cell::UnsafeCell <
        F >, $($arg : PhantomData <$type >,)* phantom_output : PhantomData < Output >, }
        impl < F, $($type,)* Output > $name < F, $($type,)* Output > where F : FnMut(&
        mut dyn $trait <$($type,)* Output >, $($type,)*) -> Output, { pub fn new(f : F)
        -> Self { Self { f : std::cell::UnsafeCell::new(f), $($arg :
        Default::default(),)* phantom_output : Default::default(), } } } impl < F,
        $($type,)* Output > $trait <$($type,)* Output > for $name < F, $($type,)* Output
        > where F : FnMut(& mut dyn $trait <$($type,)* Output >, $($type,)*) -> Output, {
        fn call(& mut self, $($arg : $type,)*) -> Output { unsafe { (* self.f.get())
        (self, $($arg,)*) } } }
    };
}
recursive_function!(RecursiveFunction0, Callable0, ());
recursive_function!(RecursiveFunction, Callable, (Arg arg,));
recursive_function!(RecursiveFunction2, Callable2, (Arg1 arg1, Arg2 arg2,));
recursive_function!(RecursiveFunction3, Callable3, (Arg1 arg1, Arg2 arg2, Arg3 arg3,));
recursive_function!(
    RecursiveFunction4, Callable4, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4,)
);
recursive_function!(
    RecursiveFunction5, Callable5, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5
    arg5,)
);
recursive_function!(
    RecursiveFunction6, Callable6, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5
    arg5, Arg6 arg6,)
);
recursive_function!(
    RecursiveFunction7, Callable7, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5
    arg5, Arg6 arg6, Arg7 arg7,)
);
recursive_function!(
    RecursiveFunction8, Callable8, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5
    arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8,)
);
recursive_function!(
    RecursiveFunction9, Callable9, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5
    arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8, Arg9 arg9,)
);
}
pub mod test_type {
pub enum TestType {
    Single,
    MultiNumber,
    MultiEof,
}
pub enum TaskType {
    Classic,
    Interactive,
}
}
}
pub mod numbers {
pub mod num_traits {
pub mod algebra {
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use std::ops::{
    Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, RemAssign, Sub, SubAssign,
};
pub trait Zero {
    fn zero() -> Self;
}
pub trait One {
    fn one() -> Self;
}
pub trait AdditionMonoid: Add<Output = Self> + AddAssign + Zero + Eq + Sized {}
impl<T: Add<Output = Self> + AddAssign + Zero + Eq> AdditionMonoid for T {}
pub trait AdditionMonoidWithSub: AdditionMonoid + Sub<Output = Self> + SubAssign {}
impl<T: AdditionMonoid + Sub<Output = Self> + SubAssign> AdditionMonoidWithSub for T {}
pub trait AdditionGroup: AdditionMonoidWithSub + Neg<Output = Self> {}
impl<T: AdditionMonoidWithSub + Neg<Output = Self>> AdditionGroup for T {}
pub trait MultiplicationMonoid: Mul<Output = Self> + MulAssign + One + Eq + Sized {}
impl<T: Mul<Output = Self> + MulAssign + One + Eq> MultiplicationMonoid for T {}
pub trait IntegerMultiplicationMonoid: MultiplicationMonoid + Div<
        Output = Self,
    > + Rem<Output = Self> + DivAssign + RemAssign {}
impl<
    T: MultiplicationMonoid + Div<Output = Self> + Rem<Output = Self> + DivAssign
        + RemAssign,
> IntegerMultiplicationMonoid for T {}
pub trait MultiplicationGroup: MultiplicationMonoid + Div<
        Output = Self,
    > + DivAssign + Invertible<Output = Self> {}
impl<
    T: MultiplicationMonoid + Div<Output = Self> + DivAssign + Invertible<Output = Self>,
> MultiplicationGroup for T {}
pub trait SemiRing: AdditionMonoid + MultiplicationMonoid {}
impl<T: AdditionMonoid + MultiplicationMonoid> SemiRing for T {}
pub trait SemiRingWithSub: AdditionMonoidWithSub + SemiRing {}
impl<T: AdditionMonoidWithSub + SemiRing> SemiRingWithSub for T {}
pub trait Ring: SemiRing + AdditionGroup {}
impl<T: SemiRing + AdditionGroup> Ring for T {}
pub trait IntegerSemiRing: SemiRing + IntegerMultiplicationMonoid {}
impl<T: SemiRing + IntegerMultiplicationMonoid> IntegerSemiRing for T {}
pub trait IntegerSemiRingWithSub: SemiRingWithSub + IntegerSemiRing {}
impl<T: SemiRingWithSub + IntegerSemiRing> IntegerSemiRingWithSub for T {}
pub trait IntegerRing: IntegerSemiRing + Ring {}
impl<T: IntegerSemiRing + Ring> IntegerRing for T {}
pub trait Field: Ring + MultiplicationGroup {}
impl<T: Ring + MultiplicationGroup> Field for T {}
macro_rules! zero_one_integer_impl {
    ($($t:ident)+) => {
        $(impl Zero for $t { fn zero() -> Self { 0 } } impl One for $t { fn one() -> Self
        { 1 } })+
    };
}
zero_one_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod invertible {
pub trait Invertible {
    type Output;
    fn inv(&self) -> Option<Self::Output>;
}
}
pub mod ord {
pub trait MinMax: PartialOrd {
    fn min_val() -> Self;
    fn max_val() -> Self;
}
macro_rules! min_max_integer_impl {
    ($($t:ident)+) => {
        $(impl MinMax for $t { fn min_val() -> Self { std::$t ::MIN } fn max_val() ->
        Self { std::$t ::MAX } })+
    };
}
min_max_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod primitive {
pub trait Primitive<T>: Copy {
    fn to(self) -> T;
    fn from(t: T) -> Self;
}
macro_rules! primitive_one {
    ($t:ident, $($u:ident)+) => {
        $(impl Primitive <$u > for $t { fn to(self) -> $u { self as $u } fn from(t : $u)
        -> Self { t as $t } })+
    };
}
macro_rules! primitive {
    ($($t:ident)+) => {
        $(primitive_one!($t, u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);)+
    };
}
primitive!(u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);
}
}
}
}