// 
pub mod solution {
//{"name":"ucup19_f","group":"Manual","url":"","interactive":false,"timeLimit":2000,"tests":[{"input":"","output":""}],"testType":"single","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"ucup19_f"}}}

use crate::algo_lib::collections::bit_set::BitSet;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::graph::topological_sort::TopologicalSort;
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::misc::recursive_function::Callable;
use crate::algo_lib::misc::recursive_function::RecursiveFunction;
use crate::algo_lib::numbers::matrix::Matrix;
use crate::algo_lib::numbers::mod_int::ModIntF;
use crate::algo_lib::numbers::num_traits::algebra::Zero;

type PreCalc = ();

fn solve(input: &mut Input, out: &mut Output, _test_case: usize, _data: &PreCalc) {
let n = input.read_size();
let q = input.read_size();

type Mod = ModIntF;

#[derive(Clone)]
enum Vertex {
None,
Internal(usize, usize),
Leaf(Mod, Mod, Mod),
Huynya(usize, Matrix<Mod>),
}

impl Readable for Vertex {
fn read(input: &mut Input) -> Self {
let tp = input.read_char();
match tp {
'x' => {
let l = input.read::<usize>() - 1;
let r = input.read::<usize>() - 1;
Vertex::Internal(l, r)
}
'v' => {
let a = input.read::<Mod>();
let b = input.read::<Mod>();
let c = input.read::<Mod>();
Vertex::Leaf(a, b, c)
}
_ => unreachable!(),
}
}
}

let mut tree = input.read_vec::<Vertex>(n);
let mut graph = Graph::new(n);
for i in 0..n {
match &tree[i] {
Vertex::Internal(l, r) => {
graph.add_edge(Edge::new(i, *l));
graph.add_edge(Edge::new(i, *r));
}
_ => {}
}
}
let mut order = graph.topological_sort().unwrap();
order.reverse();

let queries = input.read_vec::<(usize, Mod, Mod, Mod)>(q);
const BUBEN: usize = 150;

let mut condensed_tree = vec![Vertex::None; n];
let mut interesting = BitSet::new(n);
for i in (0..q).step_by(BUBEN) {
interesting.fill(false);
condensed_tree.fill(Vertex::None);
for j in i..q.min(i + BUBEN) {
interesting.set(queries[j].0 - 1);
}
for &v in &order {
interesting.set(v);
match &tree[v] {
Vertex::Internal(l, r) => {
if interesting[*l] {
if interesting[*r] {
condensed_tree[v] = Vertex::Internal(*l, *r);
} else {
let (vert, matrix) = match &condensed_tree[*l] {
Vertex::Huynya(x, y) => (*x, y.clone()),
_ => (*l, Matrix::ident(3)),
};
let (wx, wy, wz) = match &tree[*r] {
Vertex::Leaf(x, y, z) => (*x, *y, *z),
_ => unreachable!(),
};
condensed_tree[v] = Vertex::Huynya(
vert,
Matrix::new(&[
&[Mod::zero(), wz, -wy],
&[-wz, Mod::zero(), wx],
&[wy, -wx, Mod::zero()],
])
.mult(&matrix),
);
}
} else {
if interesting[*r] {
let (vert, matrix) = match &condensed_tree[*r] {
Vertex::Huynya(x, y) => (*x, y.clone()),
_ => (*r, Matrix::ident(3)),
};
let (vx, vy, vz) = match &tree[*l] {
Vertex::Leaf(x, y, z) => (*x, *y, *z),
_ => unreachable!(),
};
condensed_tree[v] = Vertex::Huynya(
vert,
Matrix::new(&[
&[Mod::zero(), -vz, vy],
&[vz, Mod::zero(), -vx],
&[-vy, vx, Mod::zero()],
])
.mult(&matrix),
);
} else {
interesting.unset(v);
let (vx, vy, vz) = match &condensed_tree[*l] {
Vertex::Leaf(x, y, z) => (*x, *y, *z),
_ => unreachable!(),
};
let (wx, wy, wz) = match &condensed_tree[*r] {
Vertex::Leaf(x, y, z) => (*x, *y, *z),
_ => unreachable!(),
};
condensed_tree[v] = Vertex::Leaf(
vy * wz - vz * wy,
vz * wx - vx * wz,
vx * wy - vy * wx,
);
}
}
}
Vertex::Leaf(a, b, c) => {
condensed_tree[v] = Vertex::Leaf(*a, *b, *c);
}
_ => unreachable!(),
}
}
for j in i..q.min(i + BUBEN) {
let (v, x, y, z) = queries[j];
condensed_tree[v - 1] = Vertex::Leaf(x, y, z);
tree[v - 1] = Vertex::Leaf(x, y, z);
let mut rec = RecursiveFunction::new(|rec, vert: usize| -> (Mod, Mod, Mod) {
match &condensed_tree[vert] {
Vertex::Leaf(a, b, c) => (*a, *b, *c),
Vertex::Huynya(v, mat) => {
let (a, b, c) = rec.call(*v);
let res = mat.mult(&Matrix::new(&[&[a], &[b], &[c]]));
(res[(0, 0)], res[(1, 0)], res[(2, 0)])
}
Vertex::Internal(l, r) => {
let (vx, vy, vz) = rec.call(*l);
let (wx, wy, wz) = rec.call(*r);
(vy * wz - vz * wy, vz * wx - vx * wz, vx * wy - vy * wx)
}
_ => unreachable!(),
}
});
out.print_line(rec.call(0));
}
}
}

pub(crate) fn run(mut input: Input, mut output: Output) -> bool {
let pre_calc = ();

#[allow(dead_code)]
enum TestType {
Single,
MultiNumber,
MultiEof,
}
let test_type = TestType::Single;
match test_type {
TestType::Single => solve(&mut input, &mut output, 1, &pre_calc),
TestType::MultiNumber => {
let t = input.read();
for i in 1..=t {
solve(&mut input, &mut output, i, &pre_calc);
}
}
TestType::MultiEof => {
let mut i = 1;
while input.peek().is_some() {
solve(&mut input, &mut output, i, &pre_calc);
i += 1;
}
}
}
output.flush();
input.skip_whitespace();
input.peek().is_none()
}

}
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;

const TRUE: bool = true;
const FALSE: bool = false;

#[derive(Clone, Eq, PartialEq)]
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 })
}

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()
}
}

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 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 dsu {
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::slice_ext::bounds::Bounds;
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use std::cell::Cell;

#[derive(Clone)]
pub struct DSU {
id: Vec<Cell<u32>>,
size: Vec<u32>,
count: usize,
}

impl DSU {
pub fn new(n: usize) -> Self {
Self {
id: (0..n).map(|i| Cell::new(i as u32)).collect_vec(),
size: vec![1; n],
count: n,
}
}

pub fn size(&self, i: usize) -> usize {
self.size[self.get(i)] as usize
}

#[allow(clippy::len_without_is_empty)]
pub fn len(&self) -> usize {
self.id.len()
}

pub fn iter(&self) -> impl Iterator<Item = usize> + '_ {
self.id.iter().enumerate().filter_map(|(i, id)| {
if (i as u32) == id.get() {
Some(i)
} else {
None
}
})
}

pub fn set_count(&self) -> usize {
self.count
}

pub fn join(&mut self, mut a: usize, mut b: usize) -> bool {
a = self.get(a);
b = self.get(b);
if a == b {
false
} else {
self.size[a] += self.size[b];
self.id[b].replace(a as u32);
self.count -= 1;
true
}
}

pub fn get(&self, i: usize) -> usize {
if self.id[i].get() != i as u32 {
let res = self.get(self.id[i].get() as usize);
self.id[i].replace(res as u32);
}
self.id[i].get() as usize
}

pub fn clear(&mut self) {
self.count = self.id.len();
self.size.legacy_fill(1);
self.id.iter().enumerate().for_each(|(i, id)| {
id.replace(i as u32);
});
}

pub fn parts(&self) -> Vec<Vec<usize>> {
let roots = self.iter().collect_vec();
let mut res = vec![Vec::new(); roots.len()];
for i in 0..self.id.len() {
res[roots.as_slice().bin_search(&self.get(i)).unwrap()].push(i);
}
res
}
}
}
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 md_arr {
pub mod arr2d {
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
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 std::ops::Index;
use std::ops::IndexMut;
use std::ops::Range;
use std::slice::Iter;
use std::vec::IntoIter;

#[derive(Clone, Eq, PartialEq, Default)]
pub struct Arr2d<T> {
d1: usize,
d2: usize,
data: Vec<T>,
}

impl<T: Clone> Arr2d<T> {
pub fn new(d1: usize, d2: usize, value: T) -> Self {
Self {
d1,
d2,
data: vec![value; d1 * d2],
}
}
}

impl<T> Arr2d<T> {
pub fn generate<F>(d1: usize, d2: usize, mut gen: F) -> Self
where
F: FnMut(usize, usize) -> T,
{
let mut data = Vec::with_capacity(d1 * d2);
for i in 0usize..d1 {
for j in 0usize..d2 {
data.push(gen(i, j));
}
}
Self { d1, d2, data }
}

pub fn d1(&self) -> usize {
self.d1
}

pub fn d2(&self) -> usize {
self.d2
}

pub fn iter(&self) -> Iter<'_, T> {
self.data.iter()
}

pub fn iter_mut(&mut self) -> impl Iterator<Item = &mut T> {
self.data.iter_mut()
}

pub fn row(&self, row: usize) -> impl Iterator<Item = &T> {
assert!(row < self.d1);
self.data.iter().skip(row * self.d2).take(self.d2)
}

pub fn row_mut(&mut self, row: usize) -> impl Iterator<Item = &mut T> {
assert!(row < self.d1);
self.data.iter_mut().skip(row * self.d2).take(self.d2)
}

pub fn column(&self, col: usize) -> impl Iterator<Item = &T> {
assert!(col < self.d2);
self.data.iter().skip(col).step_by(self.d2)
}

pub fn column_mut(&mut self, col: usize) -> impl Iterator<Item = &mut T> {
assert!(col < self.d2);
self.data.iter_mut().skip(col).step_by(self.d2)
}

pub fn swap(&mut self, r1: usize, c1: usize, r2: usize, c2: usize) {
assert!(r1 < self.d1);
assert!(r2 < self.d1);
assert!(c1 < self.d2);
assert!(c2 < self.d2);
self.data.swap(r1 * self.d2 + c1, r2 * self.d2 + c2);
}

pub fn rows(&self) -> Range<usize> {
0..self.d1
}

pub fn cols(&self) -> Range<usize> {
0..self.d2
}
}

impl<T: Clone> Arr2d<T> {
pub fn fill(&mut self, elem: T) {
self.data.legacy_fill(elem);
}

pub fn transpose(&self) -> Self {
Self::generate(self.d2, self.d1, |i, j| self[(j, i)].clone())
}
}

impl<T> Index<(usize, usize)> for Arr2d<T> {
type Output = T;

fn index(&self, (row, col): (usize, usize)) -> &Self::Output {
assert!(row < self.d1);
assert!(col < self.d2);
&self.data[self.d2 * row + col]
}
}

impl<T> Index<usize> for Arr2d<T> {
type Output = [T];

fn index(&self, index: usize) -> &Self::Output {
&self.data[self.d2 * index..self.d2 * (index + 1)]
}
}

impl<T> IndexMut<(usize, usize)> for Arr2d<T> {
fn index_mut(&mut self, (row, col): (usize, usize)) -> &mut T {
assert!(row < self.d1);
assert!(col < self.d2);
&mut self.data[self.d2 * row + col]
}
}

impl<T> IndexMut<usize> for Arr2d<T> {
fn index_mut(&mut self, index: usize) -> &mut [T] {
&mut self.data[self.d2 * index..self.d2 * (index + 1)]
}
}

impl<T> AsRef<Vec<T>> for Arr2d<T> {
fn as_ref(&self) -> &Vec<T> {
&self.data
}
}

impl<T> AsMut<Vec<T>> for Arr2d<T> {
fn as_mut(&mut self) -> &mut Vec<T> {
&mut self.data
}
}

impl<T: Writable> Writable for Arr2d<T> {
fn write(&self, output: &mut Output) {
let mut at = 0usize;
for i in 0usize..self.d1 {
if i != 0 {
output.put(b'\n');
}
for j in 0usize..self.d2 {
if j != 0 {
output.put(b' ');
}
self.data[at].write(output);
at += 1;
}
}
}
}

impl<T> IntoIterator for Arr2d<T> {
type Item = T;
type IntoIter = IntoIter<T>;

fn into_iter(self) -> Self::IntoIter {
self.data.into_iter()
}
}

impl<'a, T> IntoIterator for &'a Arr2d<T> {
type Item = &'a T;
type IntoIter = Iter<'a, T>;

fn into_iter(self) -> Self::IntoIter {
self.iter()
}
}

pub trait Arr2dRead {
fn read_table<T: Readable>(&mut self, d1: usize, d2: usize) -> Arr2d<T>;
fn read_int_table(&mut self, d1: usize, d2: usize) -> Arr2d<i32>;
fn read_long_table(&mut self, d1: usize, d2: usize) -> Arr2d<i64>;
fn read_size_table(&mut self, d1: usize, d2: usize) -> Arr2d<usize>;
fn read_char_table(&mut self, d1: usize, d2: usize) -> Arr2d<char>;
}

impl Arr2dRead for Input<'_> {
fn read_table<T: Readable>(&mut self, d1: usize, d2: usize) -> Arr2d<T> {
Arr2d::generate(d1, d2, |_, _| self.read())
}

fn read_int_table(&mut self, d1: usize, d2: usize) -> Arr2d<i32> {
self.read_table(d1, d2)
}

fn read_long_table(&mut self, d1: usize, d2: usize) -> Arr2d<i64> {
self.read_table(d1, d2)
}

fn read_size_table(&mut self, d1: usize, d2: usize) -> Arr2d<usize> {
self.read_table(d1, d2)
}

fn read_char_table(&mut self, d1: usize, d2: usize) -> Arr2d<char> {
self.read_table(d1, d2)
}
}

pub trait Arr2dCharWrite {
fn print_table(&mut self, table: &Arr2d<char>);
}

impl Arr2dCharWrite for Output<'_> {
fn print_table(&mut self, table: &Arr2d<char>) {
let mut at = 0usize;
for _ in 0..table.d1 {
for _ in 0..table.d2 {
self.print(table.data[at]);
at += 1;
}
self.put(b'\n');
}
}
}

impl<T: Readable> Readable for Arr2d<T> {
fn read(input: &mut Input) -> Self {
let d1 = input.read();
let d2 = input.read();
input.read_table(d1, d2)
}
}
}
}
pub mod slice_ext {
pub mod bounds {
pub trait Bounds<T: PartialOrd> {
fn lower_bound(&self, el: &T) -> usize;
fn upper_bound(&self, el: &T) -> usize;
fn bin_search(&self, el: &T) -> Option<usize>;
fn more(&self, el: &T) -> usize;
fn more_or_eq(&self, el: &T) -> usize;
fn less(&self, el: &T) -> usize;
fn less_or_eq(&self, el: &T) -> usize;
}

impl<T: PartialOrd> Bounds<T> for [T] {
fn lower_bound(&self, el: &T) -> usize {
let mut left = 0;
let mut right = self.len();
while left < right {
let mid = left + ((right - left) >> 1);
if &self[mid] < el {
left = mid + 1;
} else {
right = mid;
}
}
left
}

fn upper_bound(&self, el: &T) -> usize {
let mut left = 0;
let mut right = self.len();
while left < right {
let mid = left + ((right - left) >> 1);
if &self[mid] <= el {
left = mid + 1;
} else {
right = mid;
}
}
left
}

fn bin_search(&self, el: &T) -> Option<usize> {
let at = self.lower_bound(el);
if at == self.len() || &self[at] != el {
None
} else {
Some(at)
}
}

fn more(&self, el: &T) -> usize {
self.len() - self.upper_bound(el)
}

fn more_or_eq(&self, el: &T) -> usize {
self.len() - self.lower_bound(el)
}

fn less(&self, el: &T) -> usize {
self.lower_bound(el)
}

fn less_or_eq(&self, el: &T) -> usize {
self.upper_bound(el)
}
}
}
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 graph {
pub mod edges {
pub mod bi_edge {
use crate::algo_lib::graph::edges::bi_edge_trait::BiEdgeTrait;
use crate::algo_lib::graph::edges::edge_id::EdgeId;
use crate::algo_lib::graph::edges::edge_id::NoId;
use crate::algo_lib::graph::edges::edge_id::WithId;
use crate::algo_lib::graph::edges::edge_trait::BidirectionalEdgeTrait;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

#[derive(Clone)]
pub struct BiEdgeRaw<Id: EdgeId, P> {
to: u32,
id: Id,
payload: P,
}

impl<Id: EdgeId> BiEdgeRaw<Id, ()> {
pub fn new(from: usize, to: usize) -> (usize, Self) {
(
from,
Self {
to: to as u32,
id: Id::new(),
payload: (),
},
)
}
}

impl<Id: EdgeId, P> BiEdgeRaw<Id, P> {
pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
(from, Self::with_payload_impl(to, payload))
}

fn with_payload_impl(to: usize, payload: P) -> BiEdgeRaw<Id, P> {
Self {
to: to as u32,
id: Id::new(),
payload,
}
}
}

impl<Id: EdgeId, P: Clone> BidirectionalEdgeTrait for BiEdgeRaw<Id, P> {}

impl<Id: EdgeId, P: Clone> EdgeTrait for BiEdgeRaw<Id, P> {
type Payload = P;

const REVERSABLE: bool = true;

fn to(&self) -> usize {
self.to as usize
}

fn id(&self) -> usize {
self.id.id()
}

fn set_id(&mut self, id: usize) {
self.id.set_id(id);
}

fn reverse_id(&self) -> usize {
panic!("no reverse id")
}

fn set_reverse_id(&mut self, _: usize) {}

fn reverse_edge(&self, from: usize) -> Self {
Self::with_payload_impl(from, self.payload.clone())
}

fn payload(&self) -> &P {
&self.payload
}
}

impl<Id: EdgeId, P: Clone> BiEdgeTrait for BiEdgeRaw<Id, P> {}

pub type BiEdge<P> = BiEdgeRaw<NoId, P>;
pub type BiEdgeWithId<P> = BiEdgeRaw<WithId, P>;
}
pub mod bi_edge_trait {
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

pub trait BiEdgeTrait: EdgeTrait {}
}
pub mod edge {
use crate::algo_lib::graph::edges::edge_id::EdgeId;
use crate::algo_lib::graph::edges::edge_id::NoId;
use crate::algo_lib::graph::edges::edge_id::WithId;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

#[derive(Clone)]
pub struct EdgeRaw<Id: EdgeId, P> {
to: u32,
id: Id,
payload: P,
}

impl<Id: EdgeId> EdgeRaw<Id, ()> {
pub fn new(from: usize, to: usize) -> (usize, Self) {
(
from,
Self {
to: to as u32,
id: Id::new(),
payload: (),
},
)
}
}

impl<Id: EdgeId, P> EdgeRaw<Id, P> {
pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
(from, Self::with_payload_impl(to, payload))
}

fn with_payload_impl(to: usize, payload: P) -> Self {
Self {
to: to as u32,
id: Id::new(),
payload,
}
}
}

impl<Id: EdgeId, P: Clone> EdgeTrait for EdgeRaw<Id, P> {
type Payload = P;

const REVERSABLE: bool = false;

fn to(&self) -> usize {
self.to as usize
}

fn id(&self) -> usize {
self.id.id()
}

fn set_id(&mut self, id: usize) {
self.id.set_id(id);
}

fn reverse_id(&self) -> usize {
panic!("no reverse")
}

fn set_reverse_id(&mut self, _: usize) {
panic!("no reverse")
}

fn reverse_edge(&self, _: usize) -> Self {
panic!("no reverse")
}

fn payload(&self) -> &P {
&self.payload
}
}

pub type Edge<P> = EdgeRaw<NoId, P>;
pub type EdgeWithId<P> = EdgeRaw<WithId, P>;
}
pub mod edge_id {
pub trait EdgeId: Clone {
fn new() -> Self;
fn id(&self) -> usize;
fn set_id(&mut self, id: usize);
}

#[derive(Clone)]
pub struct WithId {
id: u32,
}

impl EdgeId for WithId {
fn new() -> Self {
Self { id: 0 }
}

fn id(&self) -> usize {
self.id as usize
}

fn set_id(&mut self, id: usize) {
self.id = id as u32;
}
}

#[derive(Clone)]
pub struct NoId {}

impl EdgeId for NoId {
fn new() -> Self {
Self {}
}

fn id(&self) -> usize {
panic!("Id called on no id")
}

fn set_id(&mut self, _: usize) {}
}
}
pub mod edge_trait {
pub trait EdgeTrait: Clone {
type Payload;

const REVERSABLE: bool;

fn to(&self) -> usize;
fn id(&self) -> usize;
fn set_id(&mut self, id: usize);
fn reverse_id(&self) -> usize;
fn set_reverse_id(&mut self, reverse_id: usize);
#[must_use]
fn reverse_edge(&self, from: usize) -> Self;
fn payload(&self) -> &Self::Payload;
}

pub trait BidirectionalEdgeTrait: EdgeTrait {}
}
}
pub mod graph {
use crate::algo_lib::collections::dsu::DSU;
use crate::algo_lib::graph::edges::bi_edge::BiEdge;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::edges::edge_trait::BidirectionalEdgeTrait;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use std::ops::Index;
use std::ops::IndexMut;

#[derive(Clone)]
pub struct Graph<E: EdgeTrait> {
pub(super) edges: Vec<Vec<E>>,
edge_count: usize,
}

impl<E: EdgeTrait> Graph<E> {
pub fn new(vertex_count: usize) -> Self {
Self {
edges: vec![Vec::new(); vertex_count],
edge_count: 0,
}
}

pub fn add_edge(&mut self, (from, mut edge): (usize, E)) -> usize {
let to = edge.to();
assert!(to < self.edges.len());
let direct_id = self.edges[from].len();
edge.set_id(self.edge_count);
self.edges[from].push(edge);
if E::REVERSABLE {
let rev_id = self.edges[to].len();
self.edges[from][direct_id].set_reverse_id(rev_id);
let mut rev_edge = self.edges[from][direct_id].reverse_edge(from);
rev_edge.set_id(self.edge_count);
rev_edge.set_reverse_id(direct_id);
self.edges[to].push(rev_edge);
}
self.edge_count += 1;
direct_id
}

pub fn add_vertices(&mut self, cnt: usize) {
self.edges.resize(self.edges.len() + cnt, Vec::new());
}

pub fn clear(&mut self) {
self.edge_count = 0;
for ve in self.edges.iter_mut() {
ve.clear();
}
}

pub fn vertex_count(&self) -> usize {
self.edges.len()
}

pub fn edge_count(&self) -> usize {
self.edge_count
}

pub fn degrees(&self) -> Vec<usize> {
self.edges.iter().map(|v| v.len()).collect()
}
}

impl<E: BidirectionalEdgeTrait> Graph<E> {
pub fn is_tree(&self) -> bool {
if self.edge_count + 1 != self.vertex_count() {
false
} else {
self.is_connected()
}
}

pub fn is_forest(&self) -> bool {
let mut dsu = DSU::new(self.vertex_count());
for i in 0..self.vertex_count() {
for e in self[i].iter() {
if i <= e.to() && !dsu.join(i, e.to()) {
return false;
}
}
}
true
}

pub fn is_connected(&self) -> bool {
let mut dsu = DSU::new(self.vertex_count());
for i in 0..self.vertex_count() {
for e in self[i].iter() {
dsu.join(i, e.to());
}
}
dsu.set_count() == 1
}
}

impl<E: EdgeTrait> Index<usize> for Graph<E> {
type Output = [E];

fn index(&self, index: usize) -> &Self::Output {
&self.edges[index]
}
}

impl<E: EdgeTrait> IndexMut<usize> for Graph<E> {
fn index_mut(&mut self, index: usize) -> &mut Self::Output {
&mut self.edges[index]
}
}

impl Graph<Edge<()>> {
pub fn from_edges(n: usize, edges: &[(usize, usize)]) -> Self {
let mut graph = Self::new(n);
for &(from, to) in edges {
graph.add_edge(Edge::new(from, to));
}
graph
}
}

impl Graph<BiEdge<()>> {
pub fn from_biedges(n: usize, edges: &[(usize, usize)]) -> Self {
let mut graph = Self::new(n);
for &(from, to) in edges {
graph.add_edge(BiEdge::new(from, to));
}
graph
}
}
}
pub mod topological_sort {
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use crate::algo_lib::graph::graph::Graph;
use std::collections::VecDeque;

pub trait TopologicalSort {
fn topological_sort(&self) -> Option<Vec<usize>>;
}

impl<E: EdgeTrait> TopologicalSort for Graph<E> {
fn topological_sort(&self) -> Option<Vec<usize>> {
assert!(!E::REVERSABLE);
let n = self.vertex_count();
let mut res = Vec::with_capacity(n);
let mut degree = vec![0u32; n];
for i in 0..n {
for e in self[i].iter() {
degree[e.to()] += 1;
}
}
let mut queue = VecDeque::new();
for (i, deg) in degree.iter().enumerate() {
if *deg == 0 {
queue.push_back(i);
}
}
while !queue.is_empty() {
let cur = queue.pop_front().unwrap();
res.push(cur);
for e in self[cur].iter() {
let to = e.to();
degree[to] -= 1;
if degree[to] == 0 {
queue.push_back(to);
}
}
}
if res.len() == n {
Some(res)
} else {
None
}
}
}
}
}
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,
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) -> 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, 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 !char::from(b).is_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 char::from(c).is_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) -> char {
self.skip_whitespace();
self.get().unwrap().into()
}

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 char {
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 u8 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::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]) {
for i in arg {
i.write(self);
self.put(b'\n');
}
}

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_iter_ref<'a, T: 'a + Writable, I: Iterator<Item = &'a 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 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<T: Writable> Writable for [T] {
fn write(&self, output: &mut Output) {
output.print_iter_ref(self.iter());
}
}

impl<T: Writable, const N: usize> Writable for [T; N] {
fn write(&self, output: &mut Output) {
output.print_iter_ref(self.iter());
}
}

impl<T: Writable> 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!(u8 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}

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)
}
}

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 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 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: $t: ty) => {
static mut VAL: Option<$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 = $val: expr) => {
dynamic_value!($name: $t);

$name::set_val($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 matrix {
use crate::algo_lib::collections::md_arr::arr2d::Arr2d;
use crate::algo_lib::numbers::num_traits::algebra::One;
use crate::algo_lib::numbers::num_traits::algebra::SemiRing;
use crate::algo_lib::numbers::num_traits::algebra::Zero;
use std::ops::Deref;
use std::ops::DerefMut;

#[derive(Clone)]
pub struct Matrix<T>(Arr2d<T>);

impl<T: Zero + One + Clone> Matrix<T> {
pub fn zero(n: usize, m: usize) -> Self {
Self(Arr2d::new(n, m, T::zero()))
}

pub fn ident(n: usize) -> Self {
Self(Arr2d::generate(n, n, |i, j| {
if i == j {
T::one()
} else {
T::zero()
}
}))
}
}

impl<T: Copy> Matrix<T> {
pub fn column(arr: &[T]) -> Self {
Self(Arr2d::generate(arr.len(), 1, |i, _| arr[i]))
}

pub fn row(arr: &[T]) -> Self {
Self(Arr2d::generate(1, arr.len(), |_, i| arr[i]))
}

pub fn new(arr: &[&[T]]) -> Self {
for a in arr {
assert_eq!(a.len(), arr[0].len());
}
Self(Arr2d::generate(arr.len(), arr[0].len(), |i, j| arr[i][j]))
}
}

impl<T: SemiRing + Copy> Matrix<T> {
pub fn mult(&self, a: &Matrix<T>) -> Self {
let mut res = Self::zero(self.d1(), a.d2());
Self::do_mult(&mut res, self, a);
res
}

pub fn do_mult(&mut self, a: &Matrix<T>, b: &Matrix<T>) {
assert_eq!(self.d1(), a.d1());
assert_eq!(a.d2(), b.d1());
assert_eq!(b.d2(), self.d2());
self.fill(T::zero());
for i in 0..self.d1() {
for j in 0..a.d2() {
for k in 0..b.d2() {
self[(i, k)] += a[(i, j)] * b[(j, k)];
}
}
}
}

pub fn add(&mut self, a: &Matrix<T>, b: &Matrix<T>) {
assert_eq!(self.d1(), a.d1());
assert_eq!(self.d2(), a.d2());
assert_eq!(self.d1(), b.d1());
assert_eq!(self.d2(), b.d2());
for i in 0..self.d1() {
for j in 0..self.d2() {
self[(i, j)] = a[(i, j)] + b[(i, j)];
}
}
}

pub fn add_to(&mut self, a: &Matrix<T>) {
assert_eq!(self.d1(), a.d1());
assert_eq!(self.d2(), a.d2());
for i in 0..self.d1() {
for j in 0..self.d2() {
self[(i, j)] += a[(i, j)];
}
}
}

pub fn power(&self, n: usize) -> Matrix<T> {
assert_eq!(self.d1(), self.d2());
let mut res = Self::ident(self.d1());
let mut temp = Self::ident(self.d1());
Self::do_power(self, &mut res, &mut temp, n);
res
}

fn do_power(a: &Matrix<T>, res: &mut Matrix<T>, temp: &mut Matrix<T>, n: usize) {
if n != 0 {
if (n & 1) == 0 {
Self::do_power(a, temp, res, n >> 1);
res.do_mult(temp, temp);
} else {
Self::do_power(a, temp, res, n - 1);
res.do_mult(temp, a);
}
}
}
}

impl<T> Deref for Matrix<T> {
type Target = Arr2d<T>;

fn deref(&self) -> &Self::Target {
&self.0
}
}

impl<T> DerefMut for Matrix<T> {
fn deref_mut(&mut self) -> &mut Self::Target {
&mut self.0
}
}

impl<T> From<Arr2d<T>> for Matrix<T> {
fn from(a: Arr2d<T>) -> Self {
Self(a)
}
}
}
pub mod mod_int {
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::collections::HashMap;
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 = HashMap::new();
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::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)
}

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)
}
}

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 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);
}
}
}
}
fn main() {
    let mut sin = std::io::stdin();
    let input = if false {
        algo_lib::io::input::Input::new_with_size(&mut sin, 1)
    } else {
        algo_lib::io::input::Input::new(&mut sin)
    };
    let mut stdout = std::io::stdout();
    let output = if false {
        algo_lib::io::output::Output::new_with_auto_flush(&mut stdout)
    } else {
        algo_lib::io::output::Output::new(&mut stdout)
    };
    solution::run(input, output);
}