#[allow(unused_macros)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}
 
#[allow(unused_macros)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};
 
    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}
 
#[allow(unused_macros)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };
 
    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };
 
    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };
 
    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };
 
    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };
 
    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}
 
use std::cmp::Ordering;
use std::cmp;
use std::cmp::min;
use std::collections::BTreeMap;
use std::process;
use std::cmp::Ord;
use std::collections::HashMap;
use std::collections::HashSet;
use std::collections::VecDeque;
use std::collections::BTreeSet;
use std::mem;
use std::collections::BinaryHeap;
use std::hash::{Hash, Hasher};

// ---------- begin ModInt ----------

#[derive(Clone, Copy)]
struct ModInt(usize);
impl std::ops::Add for ModInt {
    type Output = ModInt;
    fn add(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= MODu {
            d -= MODu;
        }
        ModInt(d)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: ModInt) {
        *self = *self + rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = ModInt;
    fn sub(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + MODu - rhs.0;
        if d >= MODu {
            d -= MODu;
        }
        ModInt(d)
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: ModInt) {
        *self = *self - rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = ModInt;
    fn mul(self, rhs: ModInt) -> Self::Output {
        ModInt((self.0 as u64 * rhs.0 as u64 % MODu as u64) as usize)
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: ModInt) {
        *self = *self * rhs;
    }
}

impl std::ops::Neg for ModInt {
    type Output = ModInt;
    fn neg(self) -> Self::Output {
        ModInt(if self.0 == 0 {0} else {MODu - self.0})
    }
}

impl std::fmt::Display for ModInt {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::str::FromStr for ModInt {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<usize>()?;
        Ok(ModInt::new(val))
    }
}
impl PartialEq for ModInt {
    fn eq(&self, other: &Self) -> bool {
        self.0 == other.0
    }
}
impl Hash for ModInt {
    fn hash<H: Hasher>(&self, state: &mut H) {
        self.0.hash(state);
        self.0.hash(state);
    }
}
impl Eq for ModInt {}
#[allow(dead_code)]
impl ModInt {
    pub fn new(n: usize) -> ModInt {
        ModInt(n % MODu)
    }
    pub fn zero() -> ModInt {
        ModInt(0)
    }
    pub fn one() -> ModInt {
        ModInt(1)
    }
    pub fn pow(self, mut n: usize) -> ModInt {
        let mut t = ModInt::one();
        let mut s = self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(self) -> ModInt {
        self.pow(MODu - 2)
    }
}
// ---------- end ModInt ----------

/// Equivalent to std::lowerbound and std::upperbound in c++
pub trait BinarySearch<T> {
    fn lower_bound(&self, x:&T) -> usize;
    fn upper_bound(&self, x:&T) -> usize;
}
 
impl<T: Ord> BinarySearch<T> for VecDeque<T>{
    fn lower_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();
 
        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less => {
                    low = mid + 1;
                }
                Ordering::Equal | Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
 
    fn upper_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();
 
        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less | Ordering::Equal => {
                    low = mid + 1;
                }
                Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
}
impl<T: Ord> BinarySearch<T> for [T]{
    fn lower_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();
 
        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less => {
                    low = mid + 1;
                }
                Ordering::Equal | Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
 
    fn upper_bound(&self, x: &T) -> usize {
        let mut low = 0;
        let mut high = self.len();
 
        while low != high {
            let mid = (low + high) / 2;
            match self[mid].cmp(x) {
                Ordering::Less | Ordering::Equal => {
                    low = mid + 1;
                }
                Ordering::Greater => {
                    high = mid;
                }
            }
        }
        low
    }
}

fn matmul(A:&Vec<Vec<i64>>, B:&Vec<Vec<i64>>) -> Vec<Vec<i64>>{
    let mut C = vec![vec![0;B[0].len()];A.len()];
    for i in 0..A.len(){
        for k in 0..B.len(){
            for j in 0..B[0].len(){
                C[i][j] += A[i][k]*B[k][j];
                C[i][j] %= MOD;
            }
        }
    }
    return C;
}
fn matpow(A:&mut Vec<Vec<i64>>, n:usize) -> Vec<Vec<i64>>{
    let mut B = vec![vec![0;A.len()];A.len()];
    for i in 0..A.len(){
        B[i][i] = 1;
    }
    let mut n = n;
    let mut tmp = A.clone();
    while(n>0){
        if n&1 == 1{B = matmul(&B, &tmp);}
        tmp = matmul(&tmp, &tmp);
        n>>=1;
    }
    return B;
}
fn matmul2(A:&Vec<Vec<ModInt>>, B:&Vec<Vec<ModInt>>) -> Vec<Vec<ModInt>>{
    let mut C = vec![vec![ModInt(0);B[0].len()];A.len()];
    for i in 0..A.len(){
        for k in 0..B.len(){
            for j in 0..B[0].len(){
                C[i][j] += A[i][k]*B[k][j];
            }
        }
    }
    return C;
}
fn matpow2(A:&mut Vec<Vec<ModInt>>, n:usize) -> Vec<Vec<ModInt>>{
    let mut B = vec![vec![ModInt(0);A.len()];A.len()];
    for i in 0..A.len(){
        B[i][i] = ModInt(1);
    }
    let mut n = n;
    let mut tmp = A.clone();
    while(n>0){
        if n&1 == 1{B = matmul2(&B, &tmp);}
        tmp = matmul2(&tmp, &tmp);
        n>>=1;
    }
    return B;
}
 
 
 

fn divisor(n:usize) -> Vec<usize>{
    let mut res:Vec<usize> = Vec::new(); 
    for i in 1..n+1{
        if i*i>n{break;}
        if n%i == 0{
            res.push(i);
            if i != n/i{
                res.push(n/i);    
            }    
        }    
    }
    res
}
struct UnionFind{
    par:Vec<usize>,
    rank:Vec<usize>,
    size:Vec<usize>,
    size_edge:Vec<usize>,
}
impl UnionFind{
    fn new(n:usize) -> UnionFind{
        let mut par = vec![0;n];
        for i in 0..n{
            par[i] = i;
        }
        UnionFind{
            par:par,
            rank:vec![0;n],
            size:vec![1;n],
            size_edge:vec![0;n],
        }
    }
    fn find(&mut self, x:usize) ->usize{
        if(self.par[x] == x){
            x
        }
        else{
            let p = self.par[x];
            let res = self.find(p);
            self.par[x] = res;
            res
        }
    }
    fn same(&mut self, a:usize, b:usize)->bool{
        self.find(a) == self.find(b)
    }
    fn unite(&mut self, a:usize, b:usize){
        let x = self.find(a);
        let y = self.find(b);
        if x != y{
            if (self.rank[x] < self.rank[y]){
                self.par[x] = y;
                self.size[y] += self.size[x];
                self.size_edge[y] += self.size_edge[x];
                self.size_edge[y] += 1;
            }
            else{
                self.par[y] = x;
                self.size[x] += self.size[y];
                self.size_edge[x] += self.size_edge[y];
                self.size_edge[x] += 1;
                if(self.rank[x] == self.rank[y]){ self.rank[x]+=1;}
            }
            
        }
        else{
            self.size_edge[x] += 1;
        }
    }
    fn check_size(&mut self, a:usize) -> usize{
        let x = self.find(a);
        let s = self.size[x];
        s
    }
}
pub struct Scanner<R> {
    stdin: R,
}
 
impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        std::str::from_utf8(&buf).unwrap()
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}
struct LazySegTree<BiOp> {
    n: usize,
    val: Vec<i64>,
    ma:Vec<i64>,
    op: BiOp,
    e: i64,
    upe:i64,
    inf:i64,
}

impl<BiOp> LazySegTree<BiOp>
    where BiOp: Fn(i64, i64) -> i64{
    pub fn new(n_: usize, op: BiOp, e: i64, upe:i64, inf:i64) -> Self {
        let mut n = 1;
        while n < n_ { n *= 2; } // n is a power of 2

        LazySegTree {n: n, val: vec![e; 2 * n ], ma:vec![upe;2*n], op: op, e: e, upe:upe, inf:inf}
    }
    pub fn query(&self, x:usize, y:usize, l:usize, r:usize, k:usize) -> i64 {
        if (r<=x || y<=l) {return self.inf;}
        if (x<=l && r<=y) {return self.ma[k];}
        let mut L = self.query(x,y,l,(l+r)/2, k*2);
        let mut R = self.query(x,y,(l+r)/2,r, k*2+1);
        return self.val[k] + (self.op)(L, R);

    }
    
    pub fn update(&mut self, x:usize, y:usize, v:i64, l:usize,r:usize, k:usize) {
        if (l>=r) {return;}
        if (x<=l && r<=y){
            self.val[k]+=v;
            self.ma[k]+=v;
        }
        else if(l<y && x<r){
            self.update(x, y, v, l, (l+r)/2, k*2);
            self.update(x,y,v,(l+r)/2,r, k*2+1);
            self.ma[k] = self.val[k] + (self.op)(self.ma[k*2], self.ma[k*2+1]);
        }
    }
}
fn modinv(a:ModInt)->ModInt{
    let mut a = a.0 as usize;
    let mut b = MODu as i64;
    let mut u = 1 as i64;
    let mut v = 0 as i64;
    let mut a = a as i64;
    let mut m = MODu as i64;
    while(b>0){
        let mut t = a/b;
        a -= t*b;
        mem::swap(&mut a, &mut b);
        u-=t*v;
        mem::swap(&mut u, &mut v);    
    }
    u%=m;
    if u<0{u+=m;}
    return ModInt(u as usize);

}
fn modinv2(a:usize)->usize{
    let mut a = a as usize;
    let mut b = MODu as i64;
    let mut u = 1 as i64;
    let mut v = 0 as i64;
    let mut a = a as i64;
    let mut m = MODu as i64;
    while(b>0){
        let mut t = a/b;
        a -= t*b;
        mem::swap(&mut a, &mut b);
        u-=t*v;
        mem::swap(&mut u, &mut v);
    }
    u%=m;
    if u<0{u+=m;}
    return u as usize;

}
fn modpow(x:ModInt, n:ModInt) -> ModInt{
        let mut ans = ModInt(1);
        let mut n = n.0 as usize;
        let mut x = x;
        while(n != 0){
            if (n&1 == 1){ans = ans*x;}
            x = x*x;
            n = n>>1;
        }
        ans
}
fn comb(a:usize, b:usize,  fac:&Vec<ModInt>, ifac:&Vec<ModInt>)->ModInt{
        let mut a = a;
        let mut b = b;
        if a == 0 && b == 0{return ModInt(1);}
        if a<b || a<0{return ModInt(0);}
        let mut tmp = ifac[a-b]*ifac[b];
        return tmp * fac[a];
}

fn invs()->(Vec<ModInt>, Vec<ModInt>){
    let mut fac = vec![ModInt(0);600001];
    let mut ifac = vec![ModInt(0);600001];
    fac[0] = ModInt(1);
    ifac[0] = ModInt(1);
    for i in 0..600000{

        fac[i+1] = fac[i] * ModInt(i+1);
        ifac[i+1] = ifac[i] * modpow(ModInt(i+1), ModInt(MODu - 2));
    }
    (fac, ifac)
}

struct ConvexHallTrick {
    Q: Vec<(i64, i64)>,
}

impl ConvexHallTrick{
    pub fn new() -> Self {

        ConvexHallTrick {Q: Vec::new()}
    }
    pub fn calc(&self, p:(i64, i64), x:i64)->i64{
        return p.0 * x + p.1;
    }
    pub fn dodo(& self, A:(i64, i64), B:(i64, i64), C:(i64, i64)) -> bool{
        //max or min
        (A.1 - C.1) * (B.0 - A.0) <= (A.1 - B.1)*(C.0 - A.0)
    }
    pub fn add(&mut self, a:i64, b:i64){
        self.Q.push((a, b));
        let mut v = self.Q.len();
        while(v >=3 && self.dodo(self.Q[v-3], self.Q[v-2], self.Q[v-1])){
            self.Q[v-2] = self.Q[v-1];
            self.Q.pop();
            v = self.Q.len();
        }
    }
    pub fn query(& self, x:i64) -> i64{
        let mut L = -1;
        let mut R = (self.Q.len() - 1) as i64;
        while(R-L>1){
            let mut m = (L+R)/2;
            if self.calc(self.Q[m as usize], x)>=self.calc(self.Q[m as usize+1], x){
                L=m;
            }
            else{
                R=m;
            }

        }
        return self.calc(self.Q[R as usize], x);
    }
}
#[derive(Eq, PartialEq, Clone, Debug)]
pub struct Rev<T>(pub T);
impl<T: PartialOrd> PartialOrd for Rev<T> {
    fn partial_cmp(&self, other: &Rev<T>) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }
}
impl<T: Ord> Ord for Rev<T> {
    fn cmp(&self, other: &Rev<T>) -> Ordering {
        other.0.cmp(&self.0)
    }
}
fn sieve(n:usize) -> (Vec<bool>, Vec<usize>){
    let mut p:usize = 0;
    let mut is_prime = vec![false; n+1];
    let mut prime = Vec::new();
    for i in 0..n+1{
        is_prime[i] = true;
    }
    is_prime[0] = false;
    is_prime[1] = false;
    for i in 2..n+1{
        if is_prime[i]{
            prime.push(i as usize);
            let mut j = 2*i;
            while(j<=n){
                is_prime[j] = false;
                j+=i;
            }
        }
    }
    (is_prime, prime)
    
}
fn nHr(n:usize, r:usize, fac:&Vec<ModInt>, ifac:&Vec<ModInt>) -> ModInt{
    if n+r == 0{
        return ModInt(1);
    }
    comb(n + r - 1, r, fac, ifac)
}
fn gcd(a:usize, b:usize)->usize{
    if b==0{return a;}
    return gcd(b, a%b);
}
fn lcm(a:usize, b:usize)->usize{
    return (b/gcd(a, b))*a;
}
struct SegTree_MOD<BiOp> {
    n: usize,
    dat: Vec<i64>,
    op: BiOp,
    e: i64,
    mod_:i64,
}
impl<BiOp> SegTree_MOD<BiOp>
    where BiOp: Fn(i64, i64) -> i64 
          {
    pub fn new(n_: usize, op: BiOp, e: i64, mod_:i64) -> Self {
        let mut n = 1;
        while n < n_ { n *= 2; } // n is a power of 2
        SegTree_MOD {n: n, dat: vec![e; 2 * n - 1], op: op, e: e, mod_:mod_}
    }
    /* ary[k] <- v */
    pub fn update(&mut self, idx: usize, v: i64) {
        let mut k = idx + self.n - 1;
        self.dat[k] = v;
        while k > 0 {
            k = (k - 1) / 2;
            self.dat[k] = (self.op)(self.dat[2 * k + 1], self.dat[2 * k + 2]);
            self.dat[k] %= self.mod_;
        }
    }
    /* [a, b) (note: half-inclusive)
     * http://proc-cpuinfo.fixstars.com/2017/07/optimize-segment-tree/ */
    pub fn query(&self, mut a: usize, mut b: usize) -> i64 {
        let mut left = self.e;
        let mut right = self.e;
        a += self.n - 1;
        b += self.n - 1;
        while a < b {
            if (a & 1) == 0 {
                left = (self.op)(left, self.dat[a]);
                left %= self.mod_;
            }
            if (b & 1) == 0 {
                right = (self.op)(self.dat[b - 1], right);
                right %= self.mod_;
            }
            a = a / 2;
            b = (b - 1) / 2;
        }
        let mut res = (self.op)(left, right);
        res %= self.mod_;
        res
        
    }
}
fn modpow2(x:usize, n:usize) -> usize{
        let mut ans = 1;
        let mut n = n;
        let mut x = x;
        while(n != 0){
            if (n&1 == 1){ans = ans*x%MODu;}
            x = x*x%MODu;
            n = n>>1;
        }
        ans
}
#[derive(Clone)]
struct PPUnionFind{
    par:Vec<usize>,
    rank:Vec<usize>,
    time:Vec<usize>,
    now:usize,
    history:Vec<(usize, usize)>,
}
impl PPUnionFind{
    fn new(n:usize) -> PPUnionFind{
        let mut par = vec![0;n];
        for i in 0..n{
            par[i] = i;
        }
        PPUnionFind{
            par:par,
            rank:vec![0;n],
            time:vec![INF as usize;n],
            now:0,
            history:vec![],
        }
    }
    fn find(&mut self, t:usize, x:usize) ->usize{
        if self.time[x] > t{return x;}
        else { let tt = self.par[x]; return self.find(t, tt);}
    }
    fn unite(&mut self, x:usize, y:usize) -> usize{
        self.now+=1;
        let mut x = x;
        let mut y = y;
        let nc = self.now;
        x = self.find(nc, x);
        y = self.find(nc, y);
        if x == y{return self.now;}
        if self.par[x] < self.par[y] {mem::swap(&mut x, &mut y);}
        self.par[x] += self.par[y];
        self.history.push((self.now, self.par[x]));
        self.par[y] = x;
        self.time[y] = self.now;
        return self.now;
    }
}

fn prim(cost:&Vec<Vec<(usize, i64)>>, vs:usize)->i64{
    let mut used = vec![false; vs];
    let mut bh = BinaryHeap::new();
    for j in 0..cost[0].len(){
            bh.push((cost[0][j].1 * -1, cost[0][j].0));
    }
    used[0] = true;
    let mut res = 0;
    while(bh.len()!=0){
        let mut m = bh.pop().unwrap();
        if used[m.1]{continue;}
        used[m.1] = true;
        for e in 0..cost[m.1].len(){
            if used[cost[m.1][e].0] == false{
                bh.push((cost[m.1][e].1 * -1, cost[m.1][e].0));
            }
        }
        res += m.0*-1;
    }
    return res;
}
fn kruscal(cost:&mut Vec<(i64, usize, usize)>, vs:usize)->i64{
    cost.sort();
    let mut uf = UnionFind::new(vs);
    let mut res = 0;

    for i in 0..cost.len(){
        let e = cost[i].clone();
        if uf.find(e.1) != uf.find(e.2){
            uf.unite(e.1, e.2);
            res += e.0;
        }
    }
    return res;
}
fn kruscal3(cost:&mut Vec<(f64, usize, usize, usize)>, vs:usize)->(UnionFind, Vec<usize>) {
    cost.sort_by(|a, b| (&a.0).partial_cmp(&b.0).unwrap());
    let mut uf = UnionFind::new(vs);
    let mut res = 0.0;
    let mut rv = Vec::new();
    let mut c = 0.0;
    let mut t = 0.0;
    for i in 0..cost.len(){
        let e = cost[i].clone();
        if uf.find(e.1) != uf.find(e.2){
            uf.unite(e.1, e.2);
            rv.push(e.3);
        }
    }
    return (uf,rv);
}
fn kruscal2(cost:&mut Vec<(f64, usize, usize)>, vs:usize)->f64{
    cost.sort_by(|a, b| (&a.0).partial_cmp(&b.0).unwrap());
    let mut uf = UnionFind::new(vs);
    let mut res = 0.0;
    for i in 0..cost.len(){
        let e = cost[i].clone();
        if uf.find(e.1) != uf.find(e.2){
            uf.unite(e.1, e.2);
            res+= e.0;
        }
    }
    return res;
}

struct segtree<I, Op>{
    n: usize,
    dat: Vec<I>,
    op:Op,
    e:I,
}
impl<I, Op> segtree<I, Op>


    where Op: Fn(I, I) -> I, I:Copy{
        
        pub fn new(n_:usize, op: Op, e:I)->Self{
            let mut n = 1;
            while(n<n_){n*=2;}
            segtree{n: n, dat:vec![e; 2*n-1], op:op, e:e}
        }
        pub fn update(&mut self, k:usize, a:I){
            let mut k = k;
            k += self.n-1;
            self.dat[k] = a;
            while(k>0){
                k = (k-1)/2;
                self.dat[k] = (self.op)(self.dat[k*2 + 1], self.dat[k*2+2]);
            }
        }
        pub fn query(&self, a:usize, b:usize, k:usize, l:usize, r:usize) -> I{
            if r<=a || b<=l{return self.e;}
            if a<=l && r<=b{return self.dat[k];}
            else{
                let mut vl = self.query(a, b, k*2+1, l, (l+r)/2);
                let mut vr = self.query(a, b, k*2+2, (l+r)/2, r);
                return (self.op)(vl, vr);
            }
        }
    }
struct BIT<I, Op>{
    n:usize,
    bit:Vec<I>,
    op:Op,
    e:I,
    ini:I,
}
impl <I, Op>  BIT<I, Op>
    /* 1-index*/  
    where Op: Fn(I, I) -> I, I:Copy{
        pub fn new(n_:usize, op:Op, e:I, ini:I)->Self{
            BIT{n:n_, bit:vec![e;n_+1], op:op, e:e, ini:ini}
        }
        pub fn sum(&self, i:usize)->I{
            let mut s = self.ini;
            let mut i = i as i64;
            while(i>0){
                s = (self.op)(s, self.bit[i as usize]);
                i -= i & -i;
            }
            return s;
        }
        pub fn add(&mut self, i:usize, x:I){
            let mut i = i as i64;
            while(i<=self.n as i64){
                self.bit[i as usize] = (self.op)(self.bit[i as usize], x);
                i += i & -i;
            }
        }
    }
struct Dsegtree{
    n: usize,
    datA: Vec<i64>,
    datB:Vec<i64>,
    e:i64,
}
impl Dsegtree{
        pub fn new(n_:usize, e:i64)->Self{

            Dsegtree{n:n_, datA:vec![e; 1<<22 - 1], datB:vec![e;1<<22 - 1], e:e}
        }
        pub fn update(&mut self,a:usize, b:usize, x:i64,  k:usize, l:usize, r:usize){
            //println!("{} {} {} {} {} {}", a, b, x, k , l, r);
            if a<=l && r<=b{
                self.datA[k] += x;
            }
            else if (l<b && a<r){
                self.datB[k] += (cmp::min(b, r) as i64 - cmp::max(a, l) as i64) * x;
                self.update(a, b, x, k*2+1, l, (l+r)/2);
                self.update(a, b, x, k*2+2, (l+r)/2, r);
            }
        }
        pub fn query(&self, a:usize, b:usize, k:usize, l:usize, r:usize) -> i64{
            if (b<=l || r<=a){
                return 0;
            }
            else if (a<=l && r<=b){
                return self.datA[k] * ((r as i64-l as i64)) + self.datB[k];
            }
            else{
                let mut res = (cmp::min(b, r) as i64 - cmp::max(a, l) as i64)* self.datA[k];
                res += self.query(a, b, k*2+1, l, (l+r)/2);
                res += self.query(a, b, k*2+2, (l+r)/2, r);
                return res;
            }
        }
    }
/*
unwrap_or_else
*/
fn prime_factor(n:usize)->HashMap<usize, usize>{
    let mut res = HashMap::new();
    let mut n = n;
    for i in 2..n{
        if i*i>n{break;}
        while(n%i==0){
            *res.entry(i).or_insert(0)+=1;
            n/=i;
        }
    }
    if n != 1{
        res.insert(n, 1);
    }
    res
}
struct rollinghash{
    base:Vec<i64>,
    Mod:Vec<i64>,
    hash:Vec<Vec<i64>>,
    power:Vec<Vec<i64>>,
}
impl rollinghash{
    pub fn new(s:&Vec<usize>)->Self{
        let mut n = s.len();
        let mut base = vec![1007, 2009];
        let mut hash = vec![vec![0;n+1];2];
        let mut power = vec![vec![1;n+1];2];
        let mut Mod = vec![1000000007, 1000000009];
        for iter in 0..2{
            let mut ht = vec![0;n+1];
            let mut pt = vec![1;n+1];
            for i in 0..n{
                hash[iter][i+1] = (hash[iter][i] * base[iter] + s[i] as i64) % Mod[iter];
                power[iter][i+1] = power[iter][i] * base[iter] % Mod[iter];
            }
        }
        return rollinghash{base:base, Mod:Mod, hash:hash, power:power}
   
    }
    pub fn get(&self, l:usize, r:usize)->(i64, i64){
        let mut res = self.hash[0][r] - self.hash[0][l] * self.power[0][r-l] % self.Mod[0];
        if res<0{
            res += self.Mod[0];
        }
        let mut res2 = self.hash[1][r] - self.hash[1][l] * self.power[1][r-l] % self.Mod[1];
        if res2<0{
            res2 += self.Mod[1];
        }
        return (res, res2);
    }

}
struct LCA{
    G:Vec<Vec<(usize, i64)>>,
    depth:Vec<i64>,
    len:Vec<i64>,
    parent:Vec<Vec<usize>>,
    V:usize,
    logV:usize,
}
impl LCA{
    pub fn new(V:usize, G:&Vec<Vec<(usize, i64)>>)->Self{
        let mut logV = 0;
        while(V>(1<<logV)){logV+=1;}
        return LCA{G:G.clone(), logV:logV, V:V, depth:vec![0;V], len:vec![0;V], parent:vec![vec![INFu;V];logV]}
    }
    pub fn init(&mut self, v:usize, par:usize, d:i64, l:i64){
        self.depth[v] = d;
        self.parent[0][v] = par;
        self.len[v] = l;
        for i in 0..self.G[v].len(){
            let mut w = self.G[v][i].0;
            let mut lc = self.G[v][i].1;
            if w == par{continue;}
            self.init(w, v, d+1, lc+l);
        }
    }
    pub fn build(&mut self, root:usize){
        self.init(root, INFu, 0, 0);
        for k in 0..(self.logV-1){
            for v in 0..self.V{
                if self.parent[k][v] == INFu{
                    self.parent[k+1][v] = INFu;
                }
                else{
                    self.parent[k+1][v] = self.parent[k][self.parent[k][v]];
                }
            }
        }
    }
    pub fn lca(&self, u:usize, v:usize)->usize{
        let mut u = u;
        let mut v = v;
        if self.depth[u]>self.depth[v]{
            mem::swap(&mut u, &mut v);
        }
        for k in 0..self.logV{
            if ((self.depth[v]-self.depth[u]) >> k) & 1 == 1{
                v = self.parent[k][v];
            }
        }
        if u == v{
            return u;
        }
        for k in (0..self.logV).rev(){
            if self.parent[k][u] != self.parent[k][v]{
                u = self.parent[k][u];
                v = self.parent[k][v];
            }
        }
        return self.parent[0][u];
    }
    pub fn dist(&self, u:usize, v:usize)->i64{
        let z = self.lca(u, v);
        return self.len[u]+self.len[v]-2*self.len[z];
    }
}



fn fast_prime_factor_table(ma:usize)->Vec<usize>{
    let mut p = sieve(1001);
    let mut minf = vec![0;ma];
    for j in 0..p.1.len(){
        let P = p.1[j];
        let mut now = P;
        for i in 2..ma{
            if minf[now] ==0{
                minf[now] = P;
            }
            now+=P;
            if now>=ma{
                break;
            }
        }
    }
    return minf;
}
fn area_rectanble(x1:f64, x2:f64, x3:f64, y1:f64, y2:f64, y3:f64)->f64{
    let tmp = x1*y2 + x2*y3 + x3*y1 - y1*x2 - y2*x3 - y3*x1;
    tmp.abs()/2.0
}
#[derive(PartialEq, Clone)]
struct FW(f64);

impl Eq for FW {}

impl PartialOrd for FW {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        self.0.partial_cmp(&other.0)
    }
}

impl Ord for FW {
    fn cmp(&self, other: &FW) -> Ordering {
        other.partial_cmp(self).unwrap()

    }
}
/* parsing
    fn expr(s:&Vec<char>, i:&mut usize) -> Vec<usize>{
        let mut val = term(s, i);
        while(*i<s.len() && (s[*i] == '+' || s[*i] == '-')){
            let mut op = encode(s[*i]);
            *i+=1;
            let mut val2 = term(s, i);
            val = cal(val, val2, op);
        }
        return val;
    }
    fn term(s:&Vec<char>, i:&mut usize) -> Vec<usize>{
        let mut val = factor(s, i);
        while(*i<s.len() && s[*i] == '*'){
            let mut op = encode(s[*i]);
            *i+=1;
            let mut val2 = factor(s, i);
            val = cal(val, val2, op);
        }
        return val;
    }
    fn factor(s:&Vec<char>, i:&mut usize)->Vec<usize>{
        if s[*i] == 'R' || s[*i] == 'S' || s[*i] == 'P' || s[*i] == '?'{
            let mut res = encode(s[*i]);
            *i+=1;
            return res;
        }
        *i+=1;
        let mut ret = expr(s, i);
        *i+=1;
        return ret;
    }
*/
fn matmulf64(A:&Vec<Vec<f64>>, B:&Vec<Vec<f64>>) -> Vec<Vec<f64>>{
    let mut C = vec![vec![0.0;B[0].len()];A.len()];
    for i in 0..A.len(){
        for k in 0..B.len(){
            for j in 0..B[0].len(){
                C[i][j] += A[i][k]*B[k][j];
            }
        }
    }
    return C;
}
//sort_by_key(|a| vec![a[0], -a[1]]);
//v.sort_by(|a, b| a.partial_cmp(b).unwrap());






fn fact_mod(n:usize)->usize {
    let mut  f = 1; 
    for i in 2..n+1{
        f = f * (i % MODu) % MODu;
    }
    return f;
}

fn mod_pow(x:usize, n:usize) ->usize{
    if(n == 0){ return 1;}
    let mut res = mod_pow((x * x) % MODu, n / 2 );
    if(n & 1 == 1){ res = (res * x) % MODu;}
    return res;
}

fn comb2(n:usize, r:usize)->usize {
    let mut n = n;
    let mut r = r;
    if(r > n-r) {r = n-r;}
    if(r == 0) {return 1;}
    let mut a = 1;
    for i in 0..r{
        a = a * ((n-i) % MODu) % MODu;
    }
    let mut b = mod_pow(fact_mod(r), MODu-2);
    return (a % MODu) * (b % MODu) % MODu;
}




struct scc{
    G:Vec<Vec<usize>>,
    rG:Vec<Vec<usize>>,
    vs:Vec<usize>,
    used:Vec<bool>,
    cmp:Vec<usize>,
}
impl scc{
    pub fn new(N:usize, G:&Vec<Vec<usize>>, rG:&Vec<Vec<usize>>)->Self{
        scc{G:G.clone(), rG:rG.clone(), vs:vec![], used:vec![false;N], cmp:vec![0;N]}
    }
    pub fn add_edge(&mut self, from:usize, to:usize){
        self.G[from].push(to);
        self.rG[to].push(from);
    }
    pub fn dfs(&mut self, v:usize){
        self.used[v] = true;
        for i in 0..self.G[v].len(){
            let t = self.G[v][i];
            if !self.used[t]{
                self.dfs(t);
            }
        }
        self.vs.push(v);
    }
    pub fn rdfs(&mut self, v:usize, k:usize){
        self.used[v] = true;
        self.cmp[v] = k;
        for i in 0..self.rG[v].len(){
            let t = self.rG[v][i];
            if !self.used[t]{
                self.rdfs(t, k);
            }
        }
    }
    pub fn scc(&mut self)->usize{
        for v in 0..self.G.len(){
            if !self.used[v]{
                self.dfs(v);
            }
        }
        self.used = vec![false;self.used.len()];
        let mut k = 0;
        for i in (0..self.vs.len()).rev(){
            if !self.used[self.vs[i]]{
                let t = self.vs[i];
                self.rdfs(t, k);
                k+=1;
            }
        }
        k
    }
}


#[macro_use]
fn modpow3(x:usize, n:usize,m:usize) -> usize{
        let mut ans = 1;
        let mut n = n;
        let mut x = x;
        x%=m;
        while(n != 0){
            if (n&1 == 1){ans = ans*x%m;}
            x = x*x%m;
            n = n>>1;
        }
        ans%m
}
fn cal1(x:usize, n:usize, m:usize, h:&mut HashMap<usize, usize>)->usize{
    if h.contains_key(&n){
        return h[&n];
    }
    if n == 1{
        return x;
    }
    if n == 0{
        return 0;
    }
    let mut res = 0;
    let mut a = n/2;
    let mut b = n-a;
    res = cal1(x, b, m, h)%m;
    res = ((res*modpow3((10)%m, a*x.to_string().len(), m)%m)%m + cal1(x, a, m, h)%m)%m;
    h.insert(n, res);
    return res;
}
fn cal2(x:usize, n:usize, m:usize, h:&mut HashMap<usize, usize>, l:usize)->usize{
    if h.contains_key(&n){
        return h[&n];
    }
    if n == 1{
        return x;
    }
    if n == 0{
        return 0;
    }
    let mut res = 0;
    let mut a = n/2;
    let mut b = n-a;
    res = cal2(x, b, m, h, l)%m;
    res = ((res*modpow3((10)%m, a*l, m)%m)%m + cal2(x, a, m, h, l)%m)%m;
    h.insert(n, res);
    return res;
}
fn modinv3(a:usize, m:usize)->usize{
    let mut a = a as usize;
    let mut b = m as i64;
    let mut u = 1 as i64;
    let mut v = 0 as i64;
    let mut a = a as i64;
    let mut m = m as i64;
    while(b>0){
        let mut t = a/b;
        a -= t*b;
        mem::swap(&mut a, &mut b);
        u-=t*v;
        mem::swap(&mut u, &mut v);
    }
    u%=m;
    if u<0{u+=m;}
    return u as usize;

}


fn dijkstra(s:usize, g:&Vec<Vec<(usize, i64)>>)->Vec<i64>{
    let mut dist = vec![INF;g.len()];
    dist[s] = 0;
    let mut h = BinaryHeap::new();
    h.push((0, s));
    while(h.len() != 0){
        let mut t = h.pop().unwrap();
        let mut d = t.0 * -1;
        let mut v = t.1;
        if dist[v]<d{
            continue;
        }
        for i in 0..g[v].len(){
            let n = g[v][i].0;
            let dd = g[v][i].1;
            if dist[n]>d+dd{
                dist[n] = d+dd;
                h.push((dist[n]*-1, n));
            }
        }
    }
    return dist;
}
fn dijkstra2(s:usize, g:&Vec<Vec<(i64, usize, usize)>>)->Vec<i64>{
    let mut dist = vec![INF;g.len()];
    dist[s] = 0;
    let mut h = BinaryHeap::new();
    h.push((0, s));
    while(h.len() != 0){
        let mut t = h.pop().unwrap();
        let mut d = t.0 * -1;
        let mut v = t.1;
        if dist[v]<d{
            continue;
        }
        for i in 0..g[v].len(){
            let n = g[v][i].1;
            let dd = g[v][i].0;
            if dist[n]>d+dd{
                dist[n] = d+dd;
                h.push((dist[n]*-1, n));
            }
        }
    }
    return dist;
}
struct BIT2{
    n:usize,
    bit:Vec<i64>,
    e:i64,
    ini:i64,
}
impl BIT2{
    /* 1-index*/  
        pub fn new(n_:usize, e:i64, ini:i64)->Self{
            BIT2{n:n_, bit:vec![e;n_+1], e:e, ini:ini}
        }
        pub fn sum(&self, i:usize)->i64{
            let mut s = self.ini;
            let mut i = i as i64;
            while(i>0){
                s = s+self.bit[i as usize];
                i -= i & -i;
            }
            return s;
        }
        pub fn add(&mut self, i:usize, x:i64){
            let mut i = i as i64;
            while(i<=self.n as i64){
                self.bit[i as usize] = self.bit[i as usize] + x;
                i += i & -i;
            }
        }
}
struct StarrySkyTree_min{
    n:usize,
    segMin:Vec<i64>,
    segAdd:Vec<i64>,
}
impl StarrySkyTree_min{
    pub fn new(n_:usize)->Self{
        let mut n = 1;
        while(n<n_){n*=2;}
        StarrySkyTree_min{n: n, segMin:vec![0; 2*n-1], segAdd:vec![0;2*n-1]}
    }
    pub fn add(&mut self, a:usize, b:usize, x:i64, k:usize, l:usize, r:usize){
        if (r <= a || b <= l) {return;}
        if (a <= l && r <= b){
            self.segAdd[k] += x;
            return;
        }
        self.add(a, b, x, k * 2 + 1, l, (l + r) / 2); 
        self.add(a, b, x, k * 2 + 2, (l + r) / 2, r); 
        self.segMin[k] = cmp::min(self.segMin[k * 2 + 1] + self.segAdd[k * 2 + 1], self.segMin[k * 2 + 2] + self.segAdd[k * 2 + 2]);
    }
    pub fn getMin(&self, a:usize, b:usize, k:usize, l:usize, r:usize)->i64{
        if (r <= a || b <= l){ return (INF);}
        if (a <= l && r <= b){ return(self.segMin[k] + self.segAdd[k])};
        let mut left = self.getMin(a, b, k * 2 + 1, l, (l + r) / 2); 
        let mut right = self.getMin(a, b, k * 2 + 2, (l + r) / 2, r); 
        return (cmp::min(left, right) + self.segAdd[k]); 
    }
}

struct StarrySkyTree_max{
    n:usize,
    segMin:Vec<i64>,
    segAdd:Vec<i64>,
}
impl StarrySkyTree_max{
    pub fn new(n_:usize)->Self{
        let mut n = 1;
        while(n<n_){n*=2;}
        StarrySkyTree_max{n: n, segMin:vec![0; 2*n-1], segAdd:vec![0;2*n-1]}
    }
    pub fn add(&mut self, a:usize, b:usize, x:i64, k:usize, l:usize, r:usize){
        if (r <= a || b <= l) {return;}
        if (a <= l && r <= b){
            self.segAdd[k] += x;
            return;
        }
        self.add(a, b, x, k * 2 + 1, l, (l + r) / 2);
        self.add(a, b, x, k * 2 + 2, (l + r) / 2, r);
        self.segMin[k] = cmp::max(self.segMin[k * 2 + 1] + self.segAdd[k * 2 + 1], self.segMin[k * 2 + 2] + self.segAdd[k * 2 + 2]);
    }
    pub fn getMax(&self, a:usize, b:usize, k:usize, l:usize, r:usize)->i64{
        if (r <= a || b <= l){ return (-INF);}
        if (a <= l && r <= b){ return(self.segMin[k] + self.segAdd[k])};
        let mut left = self.getMax(a, b, k * 2 + 1, l, (l + r) / 2);
        let mut right = self.getMax(a, b, k * 2 + 2, (l + r) / 2, r);
        return (cmp::max(left, right) + self.segAdd[k]);
    }
}

struct StarrySkyTree_min_with_index{
    n:usize,
    segMin:Vec<(i64, usize)>,
    segAdd:Vec<i64>,
}
impl StarrySkyTree_min_with_index{
    pub fn new(n_:usize)->Self{
        let mut n = 1;
        while(n<n_){n*=2;}
        let mut a = vec![];
        let mut b = vec![];
        for i in 0..2*n-1{
            a.push((0, 0));
            b.push(0);
        }
        let mut p = 0;
        for i in 2*n-1-n .. 2*n-1{
            a[i] = (0, p);
            if p>=n_{
                a[i] = (INF, p);
            }
            p+=1;
        }
        for i in (0..2*n-1-n).rev(){
            a[i] = a[i*2+1];
        }
        StarrySkyTree_min_with_index{n: n, segMin:a, segAdd:b}
    }
    pub fn add(&mut self, a:usize, b:usize, x:i64, k:usize, l:usize, r:usize){
        if (r <= a || b <= l) {return;}
        if (a <= l && r <= b){
            self.segAdd[k]+=x;
            return;
        }
        self.add(a, b, x, k * 2 + 1, l, (l + r) / 2);
        self.add(a, b, x, k * 2 + 2, (l + r) / 2, r);
        let mut xx = self.segMin[k*2+1].0 + self.segAdd[k*2+1];
        let mut yy = self.segMin[k*2+2].0 + self.segAdd[k*2+2];
        if xx <= yy{
            self.segMin[k] = (xx, self.segMin[k*2+1].1);
        }
        else{
            self.segMin[k] = (yy, self.segMin[k*2+2].1);
        }
    }
    pub fn getMin(&self, a:usize, b:usize, k:usize, l:usize, r:usize)->(i64, usize){
        if (r <= a || b <= l){ return (INF, self.n);}
        if (a <= l && r <= b){ return(self.segMin[k].0 + self.segAdd[k], self.segMin[k].1)};
        let mut left = self.getMin(a, b, k * 2 + 1, l, (l + r) / 2);
        let mut right = self.getMin(a, b, k * 2 + 2, (l + r) / 2, r);
        if left.0 > right.0{
            let mut pp = right.0+self.segAdd[k];
            if right.0 == INF{
                pp = INF;
            }
            return (pp, right.1);
        }
        else{
            let mut pp = left.0+self.segAdd[k];
            if left.0 == INF{
                pp = INF;
            }
            return (pp, left.1);
        }
    }
}

fn compress(N:usize, x1:&Vec<i64>,x2:&Vec<i64>, W:i64)->Vec<i64>{
    // [0,W]
    let mut xs = HashSet::new();
    for i in 0..N{
        for d in -1..2{
            let mut tx1 = x1[i]+d;
            let mut tx2 = x2[i]+d;
            if (0<=tx1 && tx1<=W){
                xs.insert(tx1);
            }
            if (0<=tx2 && tx2<=W){
                xs.insert(tx2);
            }
        }
    }
    let mut xs:Vec<i64> = xs.into_iter().collect();
    xs.sort();
    
    return xs;
}
//２次元和 [ (x, y), (x, y) )
//sum[i+1][j+1] = ((sum[i+1][j]+sum[i][j+1]+MODu-sum[i][j])%MODu+dp[i+1][j+1])%MODu;
/*for (int i = 0; i < H; i++) {

             for (int j = 0; j < W; j++) {
                 imos[i][j + 1] += imos[i][j];
             }
         }

         for (int j = 0; j < W; j++) {
             for (int i = 0; i < H; i++) {
                 imos[i + 1][j] += imos[i][j];
             }
         }

        sums[yy1][xx1]+=1;
        sums[yy2][xx2]+=1;
        sums[yy1][xx2]-=1;
        sums[yy2][xx1]-=1;
    }
    for i in 0..ys.len() {
        for j in 0..xs.len(){
            sums[i][j + 1] += sums[i][j];
            }
        }

    for j in 0..xs.len(){
        for i in 0..ys.len(){
            sums[i + 1][j] += sums[i][j];
        }
    }
*/
//3次元和 [ (x, y, z), (x, y, z) )
/*
//let mut v = vec![X1, Y1, Z1, X2, Y2, Z2];
        let mut si = vec![1i64, -1i64];
        for i in 0..2{
            for j in 0..2{
                for k in 0..2{
                    let mut X = v[0+i*3];
                    let mut Y = v[1+j*3];
                    let mut Z = v[2+k*3];
                    sum[X][Y][Z]+=si[i]*si[j]*si[k];
                }
            }
        }
    }
    for i in 0..xs.len()-1{
        for j in 0..ys.len(){
            for k in 0..zs.len(){
                sum[i+1][j][k] += sum[i][j][k];
            }
        }
    }

    for i in 0..xs.len(){
        for j in 0..ys.len()-1{
            for k in 0..zs.len(){
                sum[i][j+1][k] += sum[i][j][k];
            }
        }
    }

    for i in 0..xs.len(){
        for j in 0..ys.len(){
            for k in 0..zs.len()-1{
                sum[i][j][k+1] += sum[i][j][k];
            }
        }
    }
*/
fn safe_mod(x:i64, m:i64)->i64{
    let mut x = x;
    let mut m = m;
    x%=m;
    if x<0{
        x+=m;
    }
    return x;
}
fn inv_gcd(a:i64, b:i64)->(i64, i64){
    let mut a = a;
    let mut b = b;
    a = safe_mod(a, b);
    if (a == 0){
        return (b, 0);
    }

    let mut s = b;
    let mut t = a;
    let mut m0 = 0;
    let mut m1 = 1;

    while (t != 0) {
        let mut  u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        let mut tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0){
        m0 += b / s;
    }
    return (s, m0);
}
fn floor_sum(n: i64, m: i64, mut a: i64, mut b: i64) -> i64 {
        let mut ans = 0;
        if a >= m {
            ans += (n - 1) * n * (a / m) / 2;
            a %= m;
        }
        if b >= m {
            ans += n * (b / m);
            b %= m;
        }
 
        let y_max = (a * n + b) / m;
        let x_max = y_max * m - b;
        if y_max == 0 {
            return ans;
        }
        ans += (n - (x_max + a - 1) / a) * y_max;
        ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
        ans
    }
fn crt(r: &Vec<i64>, m: &Vec<i64>) -> (i64, i64) {
        assert_eq!(r.len(), m.len());
        let (mut r0, mut m0) = (0, 1);
        for (&(mut ri), &(mut mi)) in r.iter().zip(m.iter()) {
            assert!(1 <= mi);
            ri = safe_mod(ri, mi);
            if m0 < mi {
                mem::swap(&mut r0, &mut ri);
                mem::swap(&mut m0, &mut mi);
            }
            if m0 % mi == 0 {
                if r0 % mi != ri {
                    return (0, 0);
                }
                continue;
            }
 
            let (g, im) = inv_gcd(m0, mi);
            let u1 = mi / g;
            if (ri - r0) % g != 0 {
                return (0, 0);
            }
            let x = (ri - r0) / g % u1 * im % u1;
 
            r0 += x * m0;
            m0 *= u1; // -> lcm(m0, mi)
            if r0 < 0 {
                r0 += m0
            };
        }
 
        (r0, m0)
    }

fn solve(){
    let sssss = std::io::stdin();
    let mut sc = Scanner { stdin: sssss.lock() };
    let mut N:usize = sc.read();
    let mut K:usize = sc.read();
    let mut A = vec![0usize;N];
    
    for i in 0..N{
        A[i] = sc.read();
    }
    fn xxx(a:usize, b:usize)->usize{
        cmp::max(a, b)
    }
    let mut st = segtree::new(600010, xxx, 0);
    for i in 0..N{
        let mut bef = st.query(A[i], A[i]+1, 0, 0, st.n);
        let mut tmp = 0;
        let mut from = 0;
        if A[i]>=K{
            from = A[i]-K;
        }
        else{
            from = 0;
        }

        tmp = st.query(from, A[i]+K+1, 0, 0, st.n) + 1;
        st.update(A[i], tmp);
    }
    println!("{}", st.query(0, 300001, 0, 0, st.n));







}

fn main(){
    solve();
}
const PI:f64 = std::f64::consts::PI;
pub static MOD:i64 = 1000000007;
pub static MODu:usize = 1000000007;
pub static MODi32:i32 = 1000000007;
pub static eps:f64 = 1e-6;
const INF: i64 = 1 << 60;
const INFu:usize = 1<<62;





Measured memory_bytes: 9144