// https://judge.eluminatis-of-lu.com/contest/676ffd92569fb90008aac7da/1151
use crate::algo_lib::collections::iter_ext::iter_copied::ItersCopied;
use crate::algo_lib::collections::min_max::MinimMaxim;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::test_type::TaskType;
use crate::algo_lib::misc::test_type::TestType;
use crate::algo_lib::numbers::primes::factorize::all_divisors;
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);
    let max = a.copy_max();
    let ad: Vec<Vec<usize>> = all_divisors(max + 1, false);
    let mut cnt = vec![0; max + 1];
    for i in a {
        for j in ad[i].copy_iter() {
            cnt[j] += 1;
        }
    }
    let mut ans = vec![0; n];
    for i in 1..=max {
        if cnt[i] > 0 {
            ans[cnt[i] - 1] = i;
        }
    }
    for i in (1..n).rev() {
        let cand = ans[i];
        ans[i - 1].maxim(cand);
    }
    out.print_line(ans);
}
pub static TEST_TYPE: TestType = TestType::Single;
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 bit_set {
use crate::algo_lib::collections::iter_ext::iter_copied::ItersCopied;
use crate::algo_lib::numbers::num_traits::bit_ops::BitOps;
use std::ops::{BitAndAssign, BitOrAssign, BitXorAssign, Index, ShlAssign, ShrAssign};
const TRUE: bool = true;
const FALSE: bool = false;
#[derive(Clone, Eq, PartialEq, Hash)]
pub struct BitSet {
    data: Vec<u64>,
    len: usize,
}
impl BitSet {
    pub fn new(len: usize) -> Self {
        let data_len = if len == 0 { 0 } else { Self::index(len - 1) + 1 };
        Self {
            data: vec![0; data_len],
            len,
        }
    }
    pub fn from_slice(len: usize, set: &[usize]) -> Self {
        let mut res = Self::new(len);
        for &i in set {
            res.set(i);
        }
        res
    }
    pub fn set(&mut self, at: usize) {
        assert!(at < self.len);
        self.data[Self::index(at)].set_bit(at & 63);
    }
    pub fn unset(&mut self, at: usize) {
        assert!(at < self.len);
        self.data[Self::index(at)].unset_bit(at & 63);
    }
    pub fn change(&mut self, at: usize, value: bool) {
        if value {
            self.set(at);
        } else {
            self.unset(at);
        }
    }
    pub fn flip(&mut self, at: usize) {
        self.change(at, !self[at]);
    }
    #[allow(clippy::len_without_is_empty)]
    pub fn len(&self) -> usize {
        self.len
    }
    pub fn fill(&mut self, value: bool) {
        self.data.fill(if value { std::u64::MAX } else { 0 });
        if value {
            self.fix_last();
        }
    }
    pub fn is_superset(&self, other: &Self) -> bool {
        assert_eq!(self.len, other.len);
        for (we, them) in self.data.copy_zip(&other.data) {
            if (we & them) != them {
                return false;
            }
        }
        true
    }
    pub fn is_subset(&self, other: &Self) -> bool {
        other.is_superset(self)
    }
    pub fn iter(&self) -> BitSetIter<'_> {
        self.into_iter()
    }
    fn index(at: usize) -> usize {
        at >> 6
    }
    pub fn count_ones(&self) -> usize {
        self.data.iter().map(|x| x.count_ones() as usize).sum()
    }
    fn fix_last(&mut self) {
        if self.len & 63 != 0 {
            let mask = (1 << (self.len & 63)) - 1;
            *self.data.last_mut().unwrap() &= mask;
        }
    }
}
pub struct BitSetIter<'s> {
    at: usize,
    inside: usize,
    set: &'s BitSet,
}
impl<'s> Iterator for BitSetIter<'s> {
    type Item = usize;
    fn next(&mut self) -> Option<Self::Item> {
        while self.at < self.set.data.len()
            && (self.inside == 64 || (self.set.data[self.at] >> self.inside) == 0)
        {
            self.at += 1;
            self.inside = 0;
        }
        if self.at == self.set.data.len() {
            None
        } else {
            self.inside
                += (self.set.data[self.at] >> self.inside).trailing_zeros() as usize;
            let res = self.at * 64 + self.inside;
            self.inside += 1;
            Some(res)
        }
    }
}
impl<'a> IntoIterator for &'a BitSet {
    type Item = usize;
    type IntoIter = BitSetIter<'a>;
    fn into_iter(self) -> Self::IntoIter {
        BitSetIter {
            at: 0,
            inside: 0,
            set: self,
        }
    }
}
impl BitOrAssign<&BitSet> for BitSet {
    fn bitor_assign(&mut self, rhs: &BitSet) {
        assert_eq!(self.len, rhs.len);
        for (i, &j) in self.data.iter_mut().zip(rhs.data.iter()) {
            *i |= j;
        }
    }
}
impl BitAndAssign<&BitSet> for BitSet {
    fn bitand_assign(&mut self, rhs: &BitSet) {
        assert_eq!(self.len, rhs.len);
        for (i, &j) in self.data.iter_mut().zip(rhs.data.iter()) {
            *i &= j;
        }
    }
}
impl BitXorAssign<&BitSet> for BitSet {
    fn bitxor_assign(&mut self, rhs: &BitSet) {
        assert_eq!(self.len, rhs.len);
        for (i, &j) in self.data.iter_mut().zip(rhs.data.iter()) {
            *i ^= j;
        }
    }
}
impl ShlAssign<usize> for BitSet {
    fn shl_assign(&mut self, rhs: usize) {
        if rhs == 0 {
            return;
        }
        if rhs >= self.len {
            self.fill(false);
            return;
        }
        let small_shift = rhs & 63;
        if small_shift != 0 {
            let mut carry = 0;
            for data in self.data.iter_mut() {
                let new_carry = (*data) >> (64 - small_shift);
                *data <<= small_shift;
                *data |= carry;
                carry = new_carry;
            }
        }
        let big_shift = rhs >> 6;
        if big_shift != 0 {
            self.data.rotate_right(big_shift);
            self.data[..big_shift].fill(0);
        }
        self.fix_last();
    }
}
impl ShrAssign<usize> for BitSet {
    fn shr_assign(&mut self, rhs: usize) {
        if rhs == 0 {
            return;
        }
        if rhs >= self.len {
            self.fill(false);
            return;
        }
        let small_shift = rhs & 63;
        if small_shift != 0 {
            let mut carry = 0;
            for data in self.data.iter_mut().rev() {
                let new_carry = (*data) << (64 - small_shift);
                *data >>= small_shift;
                *data |= carry;
                carry = new_carry;
            }
        }
        let big_shift = rhs >> 6;
        if big_shift != 0 {
            self.data.rotate_left(big_shift);
            let from = self.data.len() - big_shift;
            self.data[from..].fill(0);
        }
    }
}
impl Index<usize> for BitSet {
    type Output = bool;
    fn index(&self, at: usize) -> &Self::Output {
        assert!(at < self.len);
        if self.data[Self::index(at)].is_set(at & 63) { &TRUE } else { &FALSE }
    }
}
impl From<Vec<bool>> for BitSet {
    fn from(data: Vec<bool>) -> Self {
        let mut res = Self::new(data.len());
        for (i, &value) in data.iter().enumerate() {
            res.change(i, value);
        }
        res
    }
}
}
pub mod fx_hash_map {
use std::cell::Cell;
use std::convert::TryInto;
use std::time::SystemTime;
use std::{
    collections::{HashMap, HashSet},
    hash::{BuildHasherDefault, Hasher},
    ops::BitXor,
};
pub type FxHashMap<K, V> = HashMap<K, V, BuildHasherDefault<FxHasher>>;
pub type FxHashSet<V> = HashSet<V, BuildHasherDefault<FxHasher>>;
#[derive(Default)]
pub struct FxHasher {
    hash: usize,
}
thread_local! {
    static K : Cell < usize > = Cell::new(((SystemTime::UNIX_EPOCH.elapsed().unwrap()
    .as_nanos().wrapping_mul(2) + 1) & 0xFFFFFFFFFFFFFFFF) as usize)
}
impl FxHasher {
    #[inline]
    fn add_to_hash(&mut self, i: usize) {
        self.hash = self.hash.rotate_left(5).bitxor(i).wrapping_mul(K.get());
    }
}
impl Hasher for FxHasher {
    #[inline]
    fn write(&mut self, mut bytes: &[u8]) {
        let read_usize = |bytes: &[u8]| u64::from_ne_bytes(
            bytes[..8].try_into().unwrap(),
        );
        let mut hash = FxHasher { hash: self.hash };
        while bytes.len() >= 8 {
            hash.add_to_hash(read_usize(bytes) as usize);
            bytes = &bytes[8..];
        }
        if bytes.len() >= 4 {
            hash.add_to_hash(
                u32::from_ne_bytes(bytes[..4].try_into().unwrap()) as usize,
            );
            bytes = &bytes[4..];
        }
        if bytes.len() >= 2 {
            hash.add_to_hash(
                u16::from_ne_bytes(bytes[..2].try_into().unwrap()) as usize,
            );
            bytes = &bytes[2..];
        }
        if !bytes.is_empty() {
            hash.add_to_hash(bytes[0] as usize);
        }
        self.hash = hash.hash;
    }
    #[inline]
    fn write_u8(&mut self, i: u8) {
        self.add_to_hash(i as usize);
    }
    #[inline]
    fn write_u16(&mut self, i: u16) {
        self.add_to_hash(i as usize);
    }
    #[inline]
    fn write_u32(&mut self, i: u32) {
        self.add_to_hash(i as usize);
    }
    #[inline]
    fn write_u64(&mut self, i: u64) {
        self.add_to_hash(i as usize);
    }
    #[inline]
    fn write_usize(&mut self, i: usize) {
        self.add_to_hash(i);
    }
    #[inline]
    fn finish(&self) -> u64 {
        self.hash as u64
    }
}
}
pub mod iter_ext {
pub mod iter_copied {
use std::iter::{
    Chain, Copied, Enumerate, Filter, Map, Rev, Skip, StepBy, Sum, Take, Zip,
};
pub trait ItersCopied<'a, T: 'a + Copy>: Sized + 'a
where
    &'a Self: IntoIterator<Item = &'a T>,
{
    fn copy_iter(&'a self) -> Copied<<&'a Self as IntoIterator>::IntoIter> {
        self.into_iter().copied()
    }
    fn copy_enumerate(
        &'a self,
    ) -> Enumerate<Copied<<&'a Self as IntoIterator>::IntoIter>> {
        self.copy_iter().enumerate()
    }
    fn copy_rev(&'a self) -> Rev<Copied<<&'a Self as IntoIterator>::IntoIter>>
    where
        Copied<<&'a Self as IntoIterator>::IntoIter>: DoubleEndedIterator,
    {
        self.copy_iter().rev()
    }
    fn copy_skip(
        &'a self,
        n: usize,
    ) -> Skip<Copied<<&'a Self as IntoIterator>::IntoIter>> {
        self.copy_iter().skip(n)
    }
    fn copy_take(
        &'a self,
        n: usize,
    ) -> Take<Copied<<&'a Self as IntoIterator>::IntoIter>> {
        self.copy_iter().take(n)
    }
    fn copy_chain<V>(
        &'a self,
        chained: &'a V,
    ) -> Chain<
        Copied<<&'a Self as IntoIterator>::IntoIter>,
        Copied<<&'a V as IntoIterator>::IntoIter>,
    >
    where
        &'a V: IntoIterator<Item = &'a T>,
    {
        self.copy_iter().chain(chained.into_iter().copied())
    }
    fn copy_zip<V>(
        &'a self,
        other: &'a V,
    ) -> Zip<
        Copied<<&'a Self as IntoIterator>::IntoIter>,
        Copied<<&'a V as IntoIterator>::IntoIter>,
    >
    where
        &'a V: IntoIterator<Item = &'a T>,
    {
        self.copy_iter().zip(other.into_iter().copied())
    }
    fn copy_max(&'a self) -> T
    where
        T: Ord,
    {
        self.copy_iter().max().unwrap()
    }
    fn copy_max_by_key<B, F>(&'a self, f: F) -> T
    where
        F: FnMut(&T) -> B,
        B: Ord,
    {
        self.copy_iter().max_by_key(f).unwrap()
    }
    fn copy_min(&'a self) -> T
    where
        T: Ord,
    {
        self.copy_iter().min().unwrap()
    }
    fn copy_min_by_key<B, F>(&'a self, f: F) -> T
    where
        F: FnMut(&T) -> B,
        B: Ord,
    {
        self.copy_iter().min_by_key(f).unwrap()
    }
    fn copy_sum(&'a self) -> T
    where
        T: Sum<T>,
    {
        self.copy_iter().sum()
    }
    fn copy_map<F, U>(
        &'a self,
        f: F,
    ) -> Map<Copied<<&'a Self as IntoIterator>::IntoIter>, F>
    where
        F: FnMut(T) -> U,
    {
        self.copy_iter().map(f)
    }
    fn copy_all(&'a self, f: impl FnMut(T) -> bool) -> bool {
        self.copy_iter().all(f)
    }
    fn copy_any(&'a self, f: impl FnMut(T) -> bool) -> bool {
        self.copy_iter().any(f)
    }
    fn copy_step_by(
        &'a self,
        step: usize,
    ) -> StepBy<Copied<<&'a Self as IntoIterator>::IntoIter>> {
        self.copy_iter().step_by(step)
    }
    fn copy_filter<F: FnMut(&T) -> bool>(
        &'a self,
        f: F,
    ) -> Filter<Copied<<&'a Self as IntoIterator>::IntoIter>, F> {
        self.copy_iter().filter(f)
    }
    fn copy_fold<Acc, F>(&'a self, init: Acc, f: F) -> Acc
    where
        F: FnMut(Acc, T) -> Acc,
    {
        self.copy_iter().fold(init, f)
    }
    fn copy_reduce<F>(&'a self, f: F) -> Option<T>
    where
        F: FnMut(T, T) -> T,
    {
        self.copy_iter().reduce(f)
    }
    fn copy_position<P>(&'a self, predicate: P) -> Option<usize>
    where
        P: FnMut(T) -> bool,
    {
        self.copy_iter().position(predicate)
    }
    fn copy_find(&'a self, val: T) -> Option<usize>
    where
        T: PartialEq,
    {
        self.copy_iter().position(|x| x == val)
    }
    fn copy_count(&'a self, val: T) -> usize
    where
        T: PartialEq,
    {
        self.copy_iter().filter(|&x| x == val).count()
    }
}
impl<'a, U: 'a, T: 'a + Copy> ItersCopied<'a, T> for U
where
    &'a U: IntoIterator<Item = &'a T>,
{}
}
}
pub mod min_max {
pub trait MinimMaxim<Rhs = Self>: PartialOrd + Sized {
    fn minim(&mut self, other: Rhs) -> bool;
    fn maxim(&mut self, other: Rhs) -> bool;
}
impl<T: PartialOrd> MinimMaxim for T {
    fn minim(&mut self, other: Self) -> bool {
        if other < *self {
            *self = other;
            true
        } else {
            false
        }
    }
    fn maxim(&mut self, other: Self) -> bool {
        if other > *self {
            *self = other;
            true
        } else {
            false
        }
    }
}
impl<T: PartialOrd> MinimMaxim<T> for Option<T> {
    fn minim(&mut self, other: T) -> bool {
        match self {
            None => {
                *self = Some(other);
                true
            }
            Some(v) => v.minim(other),
        }
    }
    fn maxim(&mut self, other: T) -> bool {
        match self {
            None => {
                *self = Some(other);
                true
            }
            Some(v) => v.maxim(other),
        }
    }
}
}
pub mod slice_ext {
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 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 sorted {
pub trait Sorted {
    fn sorted(self) -> Self;
}
impl<T: Ord> Sorted for Vec<T> {
    fn sorted(mut self) -> Self {
        self.sort();
        self
    }
}
}
}
}
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 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 value {
use std::hash::Hash;
pub trait Value<T>: Copy + Eq + Hash {
    fn val() -> T;
}
pub trait ConstValue<T>: Value<T> {
    const VAL: T;
}
impl<T, V: ConstValue<T>> Value<T> for V {
    fn val() -> T {
        Self::VAL
    }
}
#[macro_export]
macro_rules! value {
    ($name:ident : $t:ty = $val:expr) => {
        #[derive(Copy, Clone, Eq, PartialEq, Hash, Ord, PartialOrd, Default)] pub struct
        $name {} impl $crate ::algo_lib::misc::value::ConstValue <$t > for $name { const
        VAL : $t = $val; }
    };
}
pub trait DynamicValue<T>: Value<T> {
    fn set_val(t: T);
}
#[macro_export]
macro_rules! dynamic_value {
    ($name:ident : $t:ty) => {
        #[allow(non_upper_case_globals)] static mut $name : Option <$t > = None;
        #[derive(Copy, Clone, Eq, PartialEq, Hash, Default)] struct $name {} impl $crate
        ::algo_lib::misc::value::DynamicValue <$t > for $name { fn set_val(t : $t) {
        unsafe { $name = Some(t); } } } impl $crate ::algo_lib::misc::value::Value <$t >
        for $name { fn val() -> $t { unsafe { $name .unwrap() } } }
    };
    ($name:ident : $t:ty = $val:expr) => {
        dynamic_value!($name : $t); <$name as $crate
        ::algo_lib::misc::value::DynamicValue <$t >>::set_val($val);
    };
}
}
pub mod when {
#[macro_export]
macro_rules! when {
    {$($cond:expr => $then:expr,)*} => {
        match () { $(_ if $cond => $then,)* _ => unreachable!(), }
    };
    {$($cond:expr => $then:expr,)* else $(=>)? $else:expr $(,)?} => {
        match () { $(_ if $cond => $then,)* _ => $else, }
    };
}
}
}
pub mod numbers {
pub mod gcd {
use crate::algo_lib::numbers::num_traits::algebra::{
    IntegerMultiplicationMonoid, IntegerSemiRingWithSub, Zero,
};
use std::mem::swap;
pub fn extended_gcd<T: IntegerSemiRingWithSub + Copy>(a: T, b: T) -> (T, T, T) {
    if a == T::zero() {
        (b, T::zero(), T::one())
    } else {
        let (d, y, mut x) = extended_gcd(b % a, a);
        x -= b / a * y;
        (d, x, y)
    }
}
pub fn gcd<T: Copy + Zero + IntegerMultiplicationMonoid>(mut a: T, mut b: T) -> T {
    while b != T::zero() {
        a %= b;
        swap(&mut a, &mut b);
    }
    a
}
pub fn lcm<T: Copy + Zero + IntegerMultiplicationMonoid>(a: T, b: T) -> T {
    (a / gcd(a, b)) * b
}
pub fn remainder<T: IntegerSemiRingWithSub + Copy>(
    a1: T,
    n1: T,
    a2: T,
    n2: T,
) -> Option<T> {
    let (d, m1, m2) = extended_gcd(n1, n2);
    if (a2 - a1) % d != T::zero() {
        return None;
    }
    let m = lcm(n1, n2);
    Some(((a1 * m2) % n1 * n2 + (a2 * m1) % n2 * n1) % m)
}
}
pub mod mod_int {
use crate::algo_lib::collections::fx_hash_map::FxHashMap;
use crate::algo_lib::io::input::{Input, Readable};
use crate::algo_lib::io::output::{Output, Writable};
use crate::algo_lib::misc::value::Value;
use crate::algo_lib::numbers::gcd::extended_gcd;
use crate::algo_lib::numbers::num_traits::algebra::{Field, One, Zero};
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use crate::{value, when};
use std::fmt::{Display, Formatter};
use std::hash::Hash;
use std::marker::PhantomData;
use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};



pub trait BaseModInt<T>: Field + Copy {
    fn from(v: T) -> Self;
    fn module() -> T;
    fn value(&self) -> T;
}
macro_rules! mod_int {
    ($name:ident, $t:ty, $s:ty, $w:ty) => {
        #[derive(Copy, Clone, Eq, PartialEq, Hash, Default)] pub struct $name < V : Value
        <$t >> { n : $t, phantom : PhantomData < V >, } impl < V : Value <$t >> $name < V
        > { unsafe fn unchecked_new(n : $t) -> Self { debug_assert!(n < V::val()); Self {
        n, phantom : Default::default(), } } unsafe fn maybe_subtract_mod(mut n : $t) ->
        $t { debug_assert!(n < 2 * V::val()); if n >= V::val() { n -= V::val(); } n } }
        impl < V : Value <$t >> $name < V > { pub fn new(n : $t) -> Self { unsafe {
        Self::unchecked_new(n % (V::val())) } } pub fn new_signed(n : $s) -> Self {
        unsafe { Self::unchecked_new(Self::maybe_subtract_mod((n % (V::val() as $s) +
        V::val() as $s) as $t,)) } } pub fn val(& self) -> $t { self.n } } impl < V :
        Value <$t >> $name < V > { pub fn log(& self, alpha : Self) -> $t { let mut base
        = FxHashMap::default(); let mut exp = 0; let mut pow = Self::one(); let mut inv =
        * self; let alpha_inv = alpha.inv().unwrap(); while exp * exp < Self::module() {
        if inv == Self::one() { return exp; } base.insert(inv, exp); exp += 1; pow *=
        alpha; inv *= alpha_inv; } let step = pow; let mut i = 1; loop { if let Some(b) =
        base.get(& pow) { break exp * i + * b; } pow *= step; i += 1; } } } impl < V :
        Value <$t >> $name < V > { pub fn new_from_wide(n : $w) -> Self { unsafe {
        Self::unchecked_new(Self::maybe_subtract_mod((n % V::val() as $w + V::val() as
        $w) as $t,)) } } } impl < V : Value <$t >> Invertible for $name < V > { type
        Output = Self; fn inv(& self) -> Option < Self > { let (g, x, _) =
        extended_gcd(self.n as $w, V::val() as $w); if g != 1 { None } else {
        Some(Self::new_from_wide(x)) } } } impl < V : Value <$t >> BaseModInt <$t > for
        $name < V > { fn from(v : $t) -> Self { Self::new(v) } fn module() -> $t {
        V::val() } fn value(& self) -> $t { self.n } } impl < V : Value <$t >> From <$t >
        for $name < V > { fn from(n : $t) -> Self { Self::new(n) } } impl < V : Value <$t
        >> AddAssign for $name < V > { fn add_assign(& mut self, rhs : Self) { self.n =
        unsafe { Self::maybe_subtract_mod(self.n + rhs.n) }; } } impl < V : Value <$t >>
        Add for $name < V > { type Output = Self; fn add(mut self, rhs : Self) ->
        Self::Output { self += rhs; self } } impl < V : Value <$t >> SubAssign for $name
        < V > { fn sub_assign(& mut self, rhs : Self) { self.n = unsafe {
        Self::maybe_subtract_mod(self.n + V::val() - rhs.n) }; } } impl < V : Value <$t
        >> Sub for $name < V > { type Output = Self; fn sub(mut self, rhs : Self) ->
        Self::Output { self -= rhs; self } } impl < V : Value <$t >> MulAssign for $name
        < V > { fn mul_assign(& mut self, rhs : Self) { self.n = ((self.n as $w) * (rhs.n
        as $w) % (V::val() as $w)) as $t; } } impl < V : Value <$t >> Mul for $name < V >
        { type Output = Self; fn mul(mut self, rhs : Self) -> Self::Output { self *= rhs;
        self } } impl < V : Value <$t >> DivAssign for $name < V > {
        #[allow(clippy::suspicious_op_assign_impl)] fn div_assign(& mut self, rhs : Self)
        { * self *= rhs.inv().unwrap(); } } impl < V : Value <$t >> Div for $name < V > {
        type Output = Self; fn div(mut self, rhs : Self) -> Self::Output { self /= rhs;
        self } } impl < V : Value <$t >> Neg for $name < V > { type Output = Self; fn
        neg(mut self) -> Self::Output { if self.n != 0 { self.n = V::val() - self.n; }
        self } } impl < V : Value <$t >> Display for $name < V > { fn fmt(& self, f : &
        mut Formatter <'_ >) -> std::fmt::Result { <$t as Display >::fmt(& self.n, f) } }
        impl < V : Value <$t >> Readable for $name < V > { fn read(input : & mut Input)
        -> Self { Self::new(input.read()) } } impl < V : Value <$t >> Writable for $name
        < V > { fn write(& self, output : & mut Output) { self.n.write(output); } } impl
        < V : Value <$t >> Zero for $name < V > { fn zero() -> Self { unsafe {
        Self::unchecked_new(0) } } } impl < V : Value <$t >> One for $name < V > { fn
        one() -> Self { Self::new(1) } } impl < V : Value <$t >> std::fmt::Debug for
        $name < V > { fn fmt(& self, f : & mut Formatter) -> std::fmt::Result { let max =
        100; when! { self.n <= max => write!(f, "{}", self.n), self.n >= V::val() - max
        => write!(f, "-{}", V::val() - self.n), else => { for denominator in 1..max { for
        num in 1..max { if Self::new(num) / Self::new(denominator) == * self { return
        write!(f, "{}/{}", num, denominator); } if - Self::new(num) /
        Self::new(denominator) == * self { return write!(f, "-{}/{}", num, denominator);
        } } } write!(f, "(?? {} ??)", self.n) }, } } } impl < V : Value <$t >> AsIndex
        for $name < V > { fn from_index(idx : usize) -> Self { let v = idx as $w; if v >=
        V::val() as $w { Self::new_from_wide(v) } else { unsafe { Self::unchecked_new(v
        as $t) } } } fn to_index(self) -> usize { self.n.to_index() } }
    };
}
mod_int!(ModInt, u32, i32, i64);
mod_int!(ModInt64, u64, i64, i128);
value!(Val7 : u32 = 1_000_000_007);
pub type ModInt7 = ModInt<Val7>;
value!(Val9 : u32 = 1_000_000_009);
pub type ModInt9 = ModInt<Val9>;
value!(ValF : u32 = 998_244_353);
pub type ModIntF = ModInt<ValF>;
}
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 $t { fn one() -> Self
        { 1 } })+
    };
}
zero_one_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod as_index {
pub trait AsIndex {
    fn from_index(idx: usize) -> Self;
    fn to_index(self) -> usize;
}
macro_rules! from_index_impl {
    ($($t:ident)+) => {
        $(impl AsIndex for $t { fn from_index(idx : usize) -> Self { idx as $t } fn
        to_index(self) -> usize { self as usize } })+
    };
}
from_index_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod bit_ops {
use crate::algo_lib::numbers::num_traits::algebra::{One, Zero};
use std::ops::{
    BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Not, RangeInclusive,
    Shl, Sub,
};
use std::ops::{ShlAssign, Shr, ShrAssign};
pub trait BitOps: Copy + BitAnd<
        Output = Self,
    > + BitAndAssign + BitOr<
        Output = Self,
    > + BitOrAssign + BitXor<
        Output = Self,
    > + BitXorAssign + Not<
        Output = Self,
    > + Shl<
        usize,
        Output = Self,
    > + ShlAssign<
        usize,
    > + Shr<usize, Output = Self> + ShrAssign<usize> + Zero + One + PartialEq {
    #[inline]
    fn bit(at: usize) -> Self {
        Self::one() << at
    }
    #[inline]
    fn is_set(&self, at: usize) -> bool {
        (*self >> at & Self::one()) == Self::one()
    }
    #[inline]
    fn set_bit(&mut self, at: usize) {
        *self |= Self::bit(at);
    }
    #[inline]
    fn unset_bit(&mut self, at: usize) {
        *self &= !Self::bit(at);
    }
    #[must_use]
    #[inline]
    fn with_bit(mut self, at: usize) -> Self {
        self.set_bit(at);
        self
    }
    #[must_use]
    #[inline]
    fn without_bit(mut self, at: usize) -> Self {
        self.unset_bit(at);
        self
    }
    #[inline]
    fn flip_bit(&mut self, at: usize) {
        *self ^= Self::bit(at);
    }
    #[must_use]
    #[inline]
    fn flipped_bit(mut self, at: usize) -> Self {
        self.flip_bit(at);
        self
    }
    fn all_bits(n: usize) -> Self {
        let mut res = Self::zero();
        for i in 0..n {
            res.set_bit(i);
        }
        res
    }
    fn iter_all(n: usize) -> RangeInclusive<Self> {
        Self::zero()..=Self::all_bits(n)
    }
}
pub struct BitIter<T> {
    cur: T,
    all: T,
    ended: bool,
}
impl<T: Copy> BitIter<T> {
    pub fn new(all: T) -> Self {
        Self {
            cur: all,
            all,
            ended: false,
        }
    }
}
impl<T: BitOps + Sub<Output = T>> Iterator for BitIter<T> {
    type Item = T;
    fn next(&mut self) -> Option<Self::Item> {
        if self.ended {
            return None;
        }
        let res = self.cur;
        if self.cur == T::zero() {
            self.ended = true;
        } else {
            self.cur = (self.cur - T::one()) & self.all;
        }
        Some(res)
    }
}
impl<
    T: Copy + BitAnd<Output = Self> + BitAndAssign + BitOr<Output = Self> + BitOrAssign
        + BitXor<Output = Self> + BitXorAssign + Not<Output = Self>
        + Shl<usize, Output = Self> + ShlAssign<usize> + Shr<usize, Output = Self>
        + ShrAssign<usize> + One + Zero + PartialEq,
> BitOps for T {}
pub trait Bits: BitOps {
    fn bits() -> u32;
}
macro_rules! bits_integer_impl {
    ($($t:ident $bits:expr),+) => {
        $(impl Bits for $t { fn bits() -> u32 { $bits } })+
    };
}
bits_integer_impl!(
    i128 128, i64 64, i32 32, i16 16, i8 8, isize 64, u128 128, u64 64, u32 32, u16 16,
    u8 8, usize 64
);
}
pub mod invertible {
pub trait Invertible {
    type Output;
    fn inv(&self) -> Option<Self::Output>;
}
}
pub mod ord {
pub trait MinMax: PartialOrd {
    fn min_val() -> Self;
    fn max_val() -> Self;
}
macro_rules! min_max_integer_impl {
    ($($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);
}
}
pub mod number_ext {
use crate::algo_lib::numbers::num_traits::algebra::{
    IntegerSemiRing, MultiplicationMonoid,
};
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use std::ops::Mul;
pub trait Power {
    #[must_use]
    fn power<T: IntegerSemiRing + Copy>(&self, exp: T) -> Self;
}
impl<S: MultiplicationMonoid + Copy> Power for S {
    fn power<T: IntegerSemiRing + Copy>(&self, exp: T) -> Self {
        if exp == T::zero() {
            S::one()
        } else {
            let mut res = self.power(exp / (T::one() + T::one()));
            res *= res;
            if exp % (T::one() + T::one()) == T::one() {
                res *= *self;
            }
            res
        }
    }
}
pub fn num_digs<S: IntegerSemiRing + AsIndex + Copy>(mut copy: S) -> usize {
    let ten = S::from_index(10);
    let mut res = 0;
    while copy != S::zero() {
        copy /= ten;
        res += 1;
    }
    res
}
pub fn sum_digs<S: IntegerSemiRing + AsIndex + Copy>(mut copy: S) -> S {
    let ten = S::from_index(10);
    let mut res = S::zero();
    while copy != S::zero() {
        res += copy % ten;
        copy /= ten;
    }
    res
}
pub fn digits<S: IntegerSemiRing + AsIndex + Copy>(
    mut copy: S,
) -> impl Iterator<Item = S> {
    let ten = S::from_index(10);
    std::iter::from_fn(move || {
        if copy == S::zero() {
            None
        } else {
            let res = copy % ten;
            copy /= ten;
            Some(res)
        }
    })
}
pub trait Square {
    fn square(self) -> Self;
}
impl<T: Mul<Output = T> + Copy> Square for T {
    fn square(self) -> Self {
        self * self
    }
}
}
pub mod primes {
pub mod factorize {
use crate::algo_lib::collections::vec_ext::sorted::Sorted;
use crate::algo_lib::misc::recursive_function::{Callable2, RecursiveFunction2};
use crate::algo_lib::numbers::num_traits::algebra::MultiplicationMonoid;
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use crate::algo_lib::numbers::num_traits::primitive::Primitive;
use crate::algo_lib::numbers::primes::prime::find_divisor;
use crate::algo_lib::numbers::primes::sieve::divisor_table;
use std::cmp::Ordering;
pub trait Factorize {
    fn prime_divisors(self) -> Vec<(u64, usize)>;
    fn divisors(self) -> Vec<u64>;
    fn max_power(self, p: Self) -> usize;
}
impl<T: Primitive<u64>> Factorize for T {
    fn prime_divisors(self) -> Vec<(u64, usize)> {
        let n = self.to();
        assert!(n >= 1);
        if n == 1 {
            return Vec::new();
        }
        let d = if n > 100 {
            find_divisor(n)
        } else {
            let mut res = n;
            let mut i = 2;
            while i * i <= n {
                if n % i == 0 {
                    res = i;
                    break;
                }
                i += 1;
            }
            res
        };
        if d == n {
            return vec![(d, 1)];
        }
        let left = d.prime_divisors();
        let right = (n / d).prime_divisors();
        let mut res = Vec::new();
        let mut i = 0;
        let mut j = 0;
        while i < left.len() && j < right.len() {
            match left[i].0.cmp(&right[j].0) {
                Ordering::Less => {
                    res.push(left[i]);
                    i += 1;
                }
                Ordering::Equal => {
                    res.push((left[i].0, left[i].1 + right[j].1));
                    i += 1;
                    j += 1;
                }
                Ordering::Greater => {
                    res.push(right[j]);
                    j += 1;
                }
            }
        }
        res.extend_from_slice(&left[i..]);
        res.extend_from_slice(&right[j..]);
        res
    }
    fn divisors(self) -> Vec<u64> {
        let pd = self.prime_divisors();
        let mut res = Vec::new();
        let mut rec = RecursiveFunction2::new(|f, mut d: u64, step: usize| {
            if step == pd.len() {
                res.push(d);
            } else {
                let (p, e) = pd[step];
                for i in 0..=e {
                    f.call(d, step + 1);
                    if i < e {
                        d *= p;
                    }
                }
            }
        });
        rec.call(1, 0);
        res.sorted()
    }
    fn max_power(self, p: Self) -> usize {
        let mut res = 0;
        let mut cur = self.to();
        assert!(cur >= 1);
        let p = p.to();
        assert!(p >= 2);
        while cur % p == 0 {
            cur /= p;
            res += 1;
        }
        res
    }
}
pub fn all_divisors<T: AsIndex + PartialEq + Copy + MultiplicationMonoid + Ord>(
    n: usize,
    sorted: bool,
) -> Vec<Vec<T>> {
    let d: Vec<T> = divisor_table(n);
    let mut res = Vec::with_capacity(n);
    if n > 0 {
        res.push(Vec::new());
    }
    if n > 1 {
        res.push(vec![T::from_index(1)]);
    }
    for (i, p) in d.into_iter().enumerate().skip(2) {
        let mut q = 0;
        let mut c = i;
        while c % p.to_index() == 0 {
            c /= p.to_index();
            q += 1;
        }
        let mut cur = Vec::with_capacity(res[c].len() * (q + 1));
        let mut by = T::from_index(1);
        for j in 0..=q {
            cur.extend(res[c].iter().map(|&x| x * by));
            if j != q {
                by *= p;
            }
        }
        if sorted {
            cur.sort();
        }
        res.push(cur);
    }
    res
}
}
pub mod prime {
use crate::algo_lib::misc::random::{gen_bound, gen_range};
use crate::algo_lib::numbers::gcd::gcd;
use crate::algo_lib::numbers::mod_int::ModInt64;
use crate::algo_lib::numbers::num_traits::algebra::{One, Zero};
use crate::algo_lib::numbers::num_traits::primitive::Primitive;
use crate::algo_lib::numbers::number_ext::Power;
use crate::{dynamic_value, when};
pub fn is_prime(n: impl Primitive<u64>) -> bool {
    let n = n.to();
    if n <= 1 {
        return false;
    }
    let mut s = 0;
    let mut d = n - 1;
    while d % 2 == 0 {
        s += 1;
        d >>= 1;
    }
    if s == 0 {
        return n == 2;
    }
    dynamic_value!(IsPrimeModule : u64 = n);
    type Mod = ModInt64<IsPrimeModule>;
    for _ in 0..20 {
        let a = Mod::new(gen_bound(n));
        if a == Mod::zero() {
            continue;
        }
        if a.power(d) == Mod::one() {
            continue;
        }
        let mut dd = d;
        let mut good = true;
        for _ in 0..s {
            if a.power(dd) == -Mod::one() {
                good = false;
                break;
            }
            dd *= 2;
        }
        if good {
            return false;
        }
    }
    true
}
pub fn next_prime(mut n: u64) -> u64 {
    if n <= 2 {
        return 2;
    }
    n += 1 - (n & 1);
    while !is_prime(n) {
        n += 2;
    }
    n
}
fn brent(n: u64, x0: u64, c: u64) -> u64 {
    dynamic_value!(ModVal : u64 = n);
    type Mod = ModInt64<ModVal>;
    let mut x = Mod::new(x0);
    let c = Mod::new(c);
    let mut g = 1;
    let mut q = Mod::one();
    let mut xs = Mod::zero();
    let mut y = Mod::zero();
    let m = 128i64;
    let mut l = 1;
    while g == 1 {
        y = x;
        for _ in 1..l {
            x = x * x + c;
        }
        let mut k = 0;
        while k < l && g == 1 {
            xs = x;
            for _ in 0..m.min(l - k) {
                x = x * x + c;
                q *= y - x;
            }
            g = gcd(q.val(), n);
            k += m;
        }
        l *= 2;
    }
    if g == n {
        loop {
            xs = xs * xs + c;
            g = gcd((xs - y).val(), n);
            if g != 1 {
                break;
            }
        }
    }
    g
}
pub fn find_divisor(n: u64) -> u64 {
    when! {
        n == 1 => 1, n % 2 == 0 => 2, is_prime(n) => n, else => { loop { let res =
        brent(n, gen_range(2..n), gen_range(1..n),); if res != n { return res; } } },
    }
}
}
pub mod sieve {
use crate::algo_lib::collections::bit_set::BitSet;
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
pub fn primality_table(n: usize) -> BitSet {
    let mut res = BitSet::new(n);
    res.fill(true);
    if n > 0 {
        res.unset(0);
    }
    if n > 1 {
        res.unset(1);
    }
    let mut i = 2;
    while i * i < n {
        if res[i] {
            for j in ((i * i)..n).step_by(i) {
                res.unset(j);
            }
        }
        i += 1;
    }
    res
}
pub fn primes<T: AsIndex>(n: usize) -> Vec<T> {
    primality_table(n).into_iter().map(|i| T::from_index(i)).collect()
}
pub fn divisor_table<T: AsIndex + PartialEq>(n: usize) -> Vec<T> {
    let mut res: Vec<_> = (0..n).map(|i| T::from_index(i)).collect();
    let mut i = 2;
    while i * i < n {
        if res[i] == T::from_index(i) {
            for j in ((i * i)..n).step_by(i) {
                res[j] = T::from_index(i);
            }
        }
        i += 1;
    }
    res
}
}
}
}
}