//
pub mod solution {
//{"name":"bb7_f","group":"Manual","url":"","interactive":false,"timeLimit":2000,"tests":[{"input":"","output":""}],"testType":"multiNumber","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"bb7_f"}}}
use crate::algo_lib::collections::fenwick::FenwickTree;
use crate::algo_lib::collections::slice_ext::indices::Indices;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::BoolOutput;
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::Factorize;
type PreCalc = ();
fn solve(input: &mut Input, out: &mut Output, _test_case: usize, _data: &mut PreCalc) {
let n = input.read_size();
let x = input.read_long();
let a = input.read_long_vec(n);
let p = x.prime_divisors();
let mut qty = vec![FenwickTree::new(n); p.len()];
for i in 0..n {
for j in p.indices() {
let mut q = 0;
let mut cur = a[i];
while cur % p[j].0 == 0 {
q += 1;
cur /= p[j].0;
}
qty[j].add(i, q);
}
}
let q = input.read_size();
for _ in 0..q {
let l = input.read_size() - 1;
let r = input.read_size();
let mut ok = true;
for i in p.indices() {
if qty[i].get(l..r) < p[i].1 {
ok = false;
break;
}
}
out.print_line(ok);
}
}
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 = ();
output.set_bool_output(BoolOutput::YesNo);
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,
}
}
}
pub mod algo_lib {
pub mod collections {
pub mod bit_set {
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use crate::algo_lib::numbers::num_traits::bit_ops::BitOps;
use std::ops::BitAndAssign;
use std::ops::BitOrAssign;
use std::ops::Index;
use std::ops::ShlAssign;
use std::ops::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) {
// 1.43
self.data.legacy_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 i in 0..self.data.len() {
if self.data[i] & other.data[i] != other.data[i] {
return false;
}
}
true
}
pub fn is_subset(&self, other: &Self) -> bool {
other.is_superset(self)
}
pub fn iter(&self) -> impl Iterator<Item = usize> + '_ {
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 {
while !self.set.data[self.at].is_set(self.inside) {
self.inside += 1;
}
let res = self.at * 64 + self.inside;
if res < self.set.len {
self.inside += 1;
Some(res)
} else {
None
}
}
}
}
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 ShlAssign<usize> for BitSet {
fn shl_assign(&mut self, rhs: usize) {
if rhs == 0 {
return;
}
let small_shift = rhs & 63;
if small_shift != 0 {
let mut carry = 0;
for i in 0..self.data.len() {
let new_carry = self.data[i] >> (64 - small_shift);
self.data[i] <<= small_shift;
self.data[i] |= 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;
}
let small_shift = rhs & 63;
if small_shift != 0 {
let mut carry = 0;
for i in (0..self.data.len()).rev() {
let new_carry = self.data[i] << (64 - small_shift);
self.data[i] >>= small_shift;
self.data[i] |= 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 fenwick {
use crate::algo_lib::collections::min_max::MinimMaxim;
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;
use std::ops::RangeBounds;
#[derive(Clone)]
pub struct FenwickTree<T> {
value: Vec<T>,
}
impl<T: AdditionMonoidWithSub + Copy> FenwickTree<T> {
pub fn new(size: usize) -> Self {
Self {
value: vec![T::zero(); size],
}
}
pub fn get_to(&self, mut to: usize) -> T {
to.minim(self.value.len());
let mut result = T::zero();
while to > 0 {
to -= 1;
result += self.value[to];
to &= to + 1;
}
result
}
pub fn get(&self, bounds: impl RangeBounds<usize>) -> T {
let from = match bounds.start_bound() {
std::ops::Bound::Included(&x) => x,
std::ops::Bound::Excluded(&x) => x + 1,
std::ops::Bound::Unbounded => 0,
};
let to = match bounds.end_bound() {
std::ops::Bound::Included(&x) => x + 1,
std::ops::Bound::Excluded(&x) => x,
std::ops::Bound::Unbounded => self.value.len(),
};
if from >= to {
T::zero()
} else {
self.get_to(to) - self.get_to(from)
}
}
pub fn add(&mut self, mut at: usize, v: T) {
while at < self.value.len() {
self.value[at] += v;
at |= at + 1;
}
}
pub fn iter(&self) -> impl Iterator<Item = T> + '_ {
self.value
.iter()
.enumerate()
// edition 2021
.map(move |(i, _)| self.get(i..=i))
}
pub fn clear(&mut self) {
self.value.legacy_fill(T::zero());
}
}
impl<T: AdditionMonoidWithSub + Copy> From<&[T]> for FenwickTree<T> {
fn from(slice: &[T]) -> Self {
let mut result = Self::new(slice.len());
for (i, &v) in slice.iter().enumerate() {
result.add(i, v);
}
result
}
}
}
pub mod fx_hash_map {
// Copyright 2015 The Rust Project Developers. See the COPYRIGHT at
// http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.
use std::cell::Cell;
use std::convert::TryInto;
use std::time::SystemTime;
use std::collections::HashMap;
use std::collections::HashSet;
use std::hash::BuildHasherDefault;
use std::hash::Hasher;
use std::mem::size_of;
use std::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() >= size_of::<usize>() {
hash.add_to_hash(read_usize(bytes) as usize);
bytes = &bytes[size_of::<usize>()..];
}
if (size_of::<usize>() > 4) && (bytes.len() >= 4) {
hash.add_to_hash(u32::from_ne_bytes(bytes[..4].try_into().unwrap()) as usize);
bytes = &bytes[4..];
}
if (size_of::<usize>() > 2) && bytes.len() >= 2 {
hash.add_to_hash(u16::from_ne_bytes(bytes[..2].try_into().unwrap()) as usize);
bytes = &bytes[2..];
}
if (size_of::<usize>() > 1) && !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 collect {
pub trait IterCollect<T>: Iterator<Item = T> + Sized {
fn collect_vec(self) -> Vec<T> {
self.collect()
}
}
impl<T, I: Iterator<Item = T> + Sized> IterCollect<T> for I {}
}
}
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 legacy_fill {
// 1.50
pub trait LegacyFill<T> {
fn legacy_fill(&mut self, val: T);
}
impl<T: Clone> LegacyFill<T> for [T] {
fn legacy_fill(&mut self, val: T) {
for el in self.iter_mut() {
*el = val.clone();
}
}
}
}
}
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;
pub struct Input<'s> {
input: &'s mut (dyn Read + Send),
buf: Vec<u8>,
at: usize,
buf_read: usize,
}
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,
}
}
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,
}
}
pub fn get(&mut self) -> Option<u8> {
if self.refill_buffer() {
let res = self.buf[self.at];
self.at += 1;
if res == b'\r' {
if self.refill_buffer() && self.buf[self.at] == b'\n' {
self.at += 1;
}
return Some(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)
}
}
//noinspection RsSelfConvention
pub fn is_exhausted(&mut self) -> bool {
self.peek().is_none()
}
//noinspection RsSelfConvention
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 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)
}
}
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;
use std::io::Stderr;
use std::io::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,
}
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,
}
}
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,
}
}
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(b' ');
}
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;
}
}
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);
$(
out.put(b' ');
self.$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::primitive::Primitive;
use std::ops::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(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
}
pub fn gen(&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 next<T: Rem<Output = T> + Primitive<u64>>(&mut self, n: T) -> T
where
u64: Primitive<T>,
{
(self.gen() % n.to()).to()
}
pub fn next_bounds<T: IntegerSemiRingWithSub + Primitive<u64>>(&mut self, f: T, t: T) -> T
where
u64: Primitive<T>,
{
f + self.next(t - f + T::one())
}
}
static mut RAND: Option<Random> = None;
pub fn random() -> &'static mut Random {
unsafe {
if RAND.is_none() {
RAND = Some(Random::new(
(SystemTime::UNIX_EPOCH.elapsed().unwrap().as_nanos() & 0xFFFFFFFFFFFFFFFF) as u64,
));
}
RAND.as_mut().unwrap()
}
}
pub trait Shuffle {
fn shuffle(&mut self);
}
impl<T> Shuffle for [T] {
fn shuffle(&mut self) {
for i in self.indices() {
let at = random().next(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 = $val;
}
};
}
pub trait DynamicValue<T>: Value<T> {
//noinspection RsSelfConvention
fn set_val(t: T);
}
#[macro_export]
macro_rules! dynamic_value {
($name: ident: $t: ty, $val: ident) => {
static mut $val: Option<$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 {
$val = Some(t);
}
}
}
impl $crate::algo_lib::misc::value::Value<$t> for $name {
fn val() -> $t {
unsafe { $val.unwrap() }
}
}
};
($name: ident: $t: ty) => {
dynamic_value!($name: $t, VAL);
};
($name: ident: $t: ty = $val: expr) => {
dynamic_value!($name: $t);
$name::set_val($val);
};
($name: ident: $t: ty = $val: expr, $val_static: ident) => {
dynamic_value!($name: $t, $val_static);
$name::set_val($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;
use crate::algo_lib::numbers::num_traits::algebra::IntegerSemiRingWithSub;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::SemiRingWithSub;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use crate::algo_lib::numbers::num_traits::wideable::Wideable;
use std::mem::swap;
pub fn extended_gcd<T: IntegerSemiRingWithSub + Wideable + Copy>(a: T, b: T) -> (T, T::W, T::W)
where
T::W: Copy + SemiRingWithSub,
{
if a == T::zero() {
(b, T::W::zero(), T::W::one())
} else {
let (d, y, mut x) = extended_gcd(b % a, a);
x -= T::W::from(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 mod mod_int {
use crate::algo_lib::collections::fx_hash_map::FxHashMap;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::input::Readable;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::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;
use crate::algo_lib::numbers::num_traits::algebra::IntegerRing;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::Ring;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use crate::algo_lib::numbers::num_traits::as_index::AsIndex;
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use crate::algo_lib::numbers::num_traits::wideable::Wideable;
use crate::value;
use crate::when;
use std::fmt::Display;
use std::fmt::Formatter;
use std::hash::Hash;
use std::marker::PhantomData;
use std::ops::Add;
use std::ops::AddAssign;
use std::ops::Div;
use std::ops::DivAssign;
use std::ops::Mul;
use std::ops::MulAssign;
use std::ops::Neg;
use std::ops::Sub;
use std::ops::SubAssign;
pub trait BaseModInt: Field + Copy {
type W: IntegerRing + Copy + From<Self::T>;
type T: IntegerRing + Ord + Copy + Wideable<W = Self::W>;
fn from(v: Self::T) -> Self;
fn module() -> Self::T;
}
#[derive(Copy, Clone, Eq, PartialEq, Hash, Default)]
pub struct ModInt<T, V: Value<T>> {
n: T,
phantom: PhantomData<V>,
}
impl<T: Copy, V: Value<T>> ModInt<T, V> {
pub fn val(&self) -> T {
self.n
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> ModInt<T, V> {
unsafe fn unchecked_new(n: T) -> Self {
debug_assert!(n >= T::zero() && n < V::val());
Self {
n,
phantom: Default::default(),
}
}
unsafe fn maybe_subtract_mod(mut n: T) -> T {
debug_assert!(n < V::val() + V::val() && n >= T::zero());
if n >= V::val() {
n -= V::val();
}
n
}
}
impl<T: IntegerRing + Ord + Copy, V: Value<T>> ModInt<T, V> {
pub fn new(n: T) -> Self {
unsafe { Self::unchecked_new(Self::maybe_subtract_mod(n % (V::val()) + V::val())) }
}
}
impl<T: Copy + IntegerRing + Ord + Wideable + Hash, V: Value<T>> ModInt<T, V>
where
T::W: Copy + IntegerRing,
{
pub fn log(&self, alpha: Self) -> T {
let mut base = FxHashMap::default();
let mut exp = T::zero();
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 += T::one();
pow *= alpha;
inv *= alpha_inv;
}
let step = pow;
let mut i = T::one();
loop {
if let Some(b) = base.get(&pow) {
break exp * i + *b;
}
pow *= step;
i += T::one();
}
}
}
impl<T: Wideable + Ring + Ord + Copy, V: Value<T>> ModInt<T, V>
where
T::W: IntegerRing,
{
pub fn new_from_wide(n: T::W) -> Self {
unsafe {
Self::unchecked_new(Self::maybe_subtract_mod(
T::downcast(n % V::val().into()) + V::val(),
))
}
}
}
impl<T: Copy + IntegerRing + Ord + Wideable, V: Value<T>> Invertible for ModInt<T, V>
where
T::W: Copy + IntegerRing,
{
type Output = Self;
fn inv(&self) -> Option<Self> {
let (g, x, _) = extended_gcd(self.n, V::val());
if g != T::one() {
None
} else {
Some(Self::new_from_wide(x))
}
}
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> BaseModInt for ModInt<T, V>
where
T::W: IntegerRing + Copy,
{
type W = T::W;
type T = T;
fn from(v: Self::T) -> Self {
Self::new(v)
}
fn module() -> T {
V::val()
}
}
impl<T: IntegerRing + Ord + Copy, V: Value<T>> From<T> for ModInt<T, V> {
fn from(n: T) -> Self {
Self::new(n)
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> AddAssign for ModInt<T, V> {
fn add_assign(&mut self, rhs: Self) {
self.n = unsafe { Self::maybe_subtract_mod(self.n + rhs.n) };
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> Add for ModInt<T, V> {
type Output = Self;
fn add(mut self, rhs: Self) -> Self::Output {
self += rhs;
self
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> SubAssign for ModInt<T, V> {
fn sub_assign(&mut self, rhs: Self) {
self.n = unsafe { Self::maybe_subtract_mod(self.n + V::val() - rhs.n) };
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> Sub for ModInt<T, V> {
type Output = Self;
fn sub(mut self, rhs: Self) -> Self::Output {
self -= rhs;
self
}
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> MulAssign for ModInt<T, V>
where
T::W: IntegerRing + Copy,
{
fn mul_assign(&mut self, rhs: Self) {
self.n = T::downcast(T::W::from(self.n) * T::W::from(rhs.n) % T::W::from(V::val()));
}
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> Mul for ModInt<T, V>
where
T::W: IntegerRing + Copy,
{
type Output = Self;
fn mul(mut self, rhs: Self) -> Self::Output {
self *= rhs;
self
}
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> DivAssign for ModInt<T, V>
where
T::W: IntegerRing + Copy,
{
#[allow(clippy::suspicious_op_assign_impl)]
fn div_assign(&mut self, rhs: Self) {
*self *= rhs.inv().unwrap();
}
}
impl<T: IntegerRing + Ord + Copy + Wideable, V: Value<T>> Div for ModInt<T, V>
where
T::W: IntegerRing + Copy,
{
type Output = Self;
fn div(mut self, rhs: Self) -> Self::Output {
self /= rhs;
self
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> Neg for ModInt<T, V> {
type Output = Self;
fn neg(mut self) -> Self::Output {
self.n = unsafe { Self::maybe_subtract_mod(V::val() - self.n) };
self
}
}
impl<T: Display, V: Value<T>> Display for ModInt<T, V> {
fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
<T as Display>::fmt(&self.n, f)
}
}
impl<T: IntegerRing + Ord + Copy + Readable, V: Value<T>> Readable for ModInt<T, V> {
fn read(input: &mut Input) -> Self {
Self::new(T::read(input))
}
}
impl<T: Writable, V: Value<T>> Writable for ModInt<T, V> {
fn write(&self, output: &mut Output) {
self.n.write(output);
}
}
impl<T: Ring + Ord + Copy, V: Value<T>> Zero for ModInt<T, V> {
fn zero() -> Self {
unsafe { Self::unchecked_new(T::zero()) }
}
}
impl<T: IntegerRing + Ord + Copy, V: Value<T>> One for ModInt<T, V> {
fn one() -> Self {
Self::new(T::one())
}
}
impl<T, V: Value<T>> Wideable for ModInt<T, V> {
type W = Self;
fn downcast(w: Self::W) -> Self {
w
}
}
impl<T: IntegerRing + Ord + Copy + Wideable + Display + AsIndex, V: Value<T>> std::fmt::Debug
for ModInt<T, V>
where
T::W: IntegerRing + Copy,
{
fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
let max = T::from_index(100);
when! {
self.n <= max => write!(f, "{}", self.n),
self.n >= V::val() - max => write!(f, "{}", self.n - V::val()),
else => {
let mut denominator = T::one();
while denominator < max {
let mut num = T::one();
while num < 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);
}
num += T::one();
}
denominator += T::one();
}
write!(f, "(?? {} ??)", self.n)
},
}
}
}
impl<T: IntegerRing + Ord + Copy + AsIndex, V: Value<T>> AsIndex for ModInt<T, V> {
fn from_index(idx: usize) -> Self {
Self::new(T::from_index(idx))
}
fn to_index(self) -> usize {
self.n.to_index()
}
}
value!(Val7: i32 = 1_000_000_007);
pub type ModInt7 = ModInt<i32, Val7>;
value!(Val9: i32 = 1_000_000_009);
pub type ModInt9 = ModInt<i32, Val9>;
value!(ValF: i32 = 998_244_353);
pub type ModIntF = ModInt<i32, ValF>;
}
pub mod num_traits {
pub mod algebra {
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
use std::ops::Add;
use std::ops::AddAssign;
use std::ops::Div;
use std::ops::DivAssign;
use std::ops::Mul;
use std::ops::MulAssign;
use std::ops::Neg;
use std::ops::Rem;
use std::ops::RemAssign;
use std::ops::Sub;
use std::ops::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;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use std::ops::BitAnd;
use std::ops::BitAndAssign;
use std::ops::BitOr;
use std::ops::BitOrAssign;
use std::ops::BitXor;
use std::ops::BitXorAssign;
use std::ops::Not;
use std::ops::RangeInclusive;
use std::ops::Shl;
use std::ops::Sub;
use std::ops::ShlAssign;
use std::ops::Shr;
use std::ops::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
{
fn bit(at: usize) -> Self {
Self::one() << at
}
fn is_set(&self, at: usize) -> bool {
(*self >> at & Self::one()) == Self::one()
}
fn set_bit(&mut self, at: usize) {
*self |= Self::bit(at)
}
fn unset_bit(&mut self, at: usize) {
*self &= !Self::bit(at)
}
#[must_use]
fn with_bit(mut self, at: usize) -> Self {
self.set_bit(at);
self
}
#[must_use]
fn without_bit(mut self, at: usize) -> Self {
self.unset_bit(at);
self
}
fn flip_bit(&mut self, at: usize) {
*self ^= Self::bit(at)
}
#[must_use]
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 {
$bits
}
}
)+};
}
bits_integer_impl!(i128 128, i64 64, i32 32, i16 16, i8 8, isize 64, u128 128, u64 64, u32 32, u16 16, u8 8, usize 64);
}
pub mod invertible {
pub trait Invertible {
type Output;
fn inv(&self) -> Option<Self::Output>;
}
}
pub mod primitive {
pub trait Primitive<T>: Copy {
fn to(self) -> T;
}
macro_rules! primitive_one {
($t: ident, $($u: ident)+) => {$(
impl Primitive<$u> for $t {
fn to(self) -> $u {
self as $u
}
}
)+};
}
macro_rules! primitive {
($($t: ident)+) => {$(
primitive_one!($t, u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);
)+}
}
primitive!(u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);
}
pub mod wideable {
use std::convert::From;
pub trait Wideable: Sized {
type W: From<Self>;
fn downcast(w: Self::W) -> Self;
}
macro_rules! wideable_impl {
($($t: ident $w: ident),+) => {$(
impl Wideable for $t {
type W = $w;
fn downcast(w: Self::W) -> Self {
w as $t
}
}
)+};
}
wideable_impl!(i64 i128, i32 i64, i16 i32, i8 i16, u64 u128, u32 u64, u16 u32, u8 u16);
}
}
pub mod number_ext {
use crate::algo_lib::numbers::num_traits::algebra::IntegerSemiRing;
use crate::algo_lib::numbers::num_traits::algebra::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 trait NumDigs {
fn num_digs(&self) -> usize;
}
impl<S: IntegerSemiRing + AsIndex + Copy> NumDigs for S {
fn num_digs(&self) -> usize {
let mut copy = *self;
let ten = S::from_index(10);
let mut res = 0;
while copy != S::zero() {
copy /= ten;
res += 1;
}
res
}
}
pub trait SumDigs {
fn sum_digs(&self) -> Self;
}
impl<S: IntegerSemiRing + AsIndex + Copy> SumDigs for S {
fn sum_digs(&self) -> S {
let mut copy = *self;
let ten = S::from_index(10);
let mut res = S::zero();
while copy != S::zero() {
res += copy % ten;
copy /= ten;
}
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;
use crate::algo_lib::misc::recursive_function::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<(i64, usize)>;
fn divisors(self) -> Vec<i64>;
fn max_power(self, p: Self) -> usize;
}
impl<T: Primitive<i64>> Factorize for T {
fn prime_divisors(self) -> Vec<(i64, 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<i64> {
let pd = self.prime_divisors();
let mut res = Vec::new();
let mut rec = RecursiveFunction2::new(|f, mut d: i64, 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::random;
use crate::algo_lib::misc::value::DynamicValue;
use crate::algo_lib::numbers::gcd::gcd;
use crate::algo_lib::numbers::mod_int::ModInt;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use crate::algo_lib::numbers::num_traits::primitive::Primitive;
use crate::algo_lib::numbers::number_ext::Power;
use crate::dynamic_value;
use crate::when;
pub fn is_prime(n: impl Primitive<i64>) -> 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: i64 = n);
type Mod = ModInt<i64, IsPrimeModule>;
for _ in 0..20 {
let a = Mod::new(random().next(n as u64) as i64);
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: i64) -> i64 {
if n <= 2 {
return 2;
}
n += 1 - (n & 1);
while !is_prime(n) {
n += 2;
}
n
}
fn brent(n: i64, x0: i64, c: i64) -> i64 {
dynamic_value!(ModVal: i64 = n);
type Mod = ModInt<i64, 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: i64) -> i64 {
when! {
n == 1 => 1,
n % 2 == 0 => 2,
is_prime(n) => n,
else => {
loop {
let res = brent(
n,
random().next_bounds(2, n as u64 - 1) as i64,
random().next_bounds(1, n as u64 - 1) as i64,
);
if res != n {
return res;
}
}
},
}
}
}
pub mod sieve {
use crate::algo_lib::collections::bit_set::BitSet;
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
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_vec()
}
pub fn divisor_table<T: AsIndex + PartialEq>(n: usize) -> Vec<T> {
let mut res = (0..n).map(|i| T::from_index(i)).collect_vec();
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
}
}
}
}
}
fn main() {
let mut sin = std::io::stdin();
let input = algo_lib::io::input::Input::new(&mut sin);
let mut stdout = std::io::stdout();
let output = algo_lib::io::output::Output::new(&mut stdout);
solution::run(input, output);
}
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