// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($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)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr,) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, ( $($t:tt),* )) => { ($(read_value!($next, $t)),*) }; ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::>() }; ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl Default for ModInt { fn default() -> Self { Self::new_internal(0) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl ::std::fmt::Debug for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { let (mut a, mut b, _) = red(self.x, M::m()); if b < 0 { a = -a; b = -b; } write!(f, "{}/{}", a, b) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } // Finds the simplest fraction x/y congruent to r mod p. // The return value (x, y, z) satisfies x = y * r + z * p. fn red(r: i64, p: i64) -> (i64, i64, i64) { if r.abs() <= 10000 { return (r, 1, 0); } let mut nxt_r = p % r; let mut q = p / r; if 2 * nxt_r >= r { nxt_r -= r; q += 1; } if 2 * nxt_r <= -r { nxt_r += r; q -= 1; } let (x, z, y) = red(nxt_r, r); (x, y - q * z, z) } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] pub struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 998_244_353; define_mod!(P, MOD); type MInt = mod_int::ModInt

; // FFT (in-place, verified as NTT only) // R: Ring + Copy // Verified by: https://judge.yosupo.jp/submission/53831 // Adopts the technique used in https://judge.yosupo.jp/submission/3153. mod fft { use std::ops::*; // n should be a power of 2. zeta is a primitive n-th root of unity. // one is unity // Note that the result is bit-reversed. pub fn fft(f: &mut [R], zeta: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let mut m = n; let mut base = zeta; unsafe { while m > 2 { m >>= 1; let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m); *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = w * (u - d); w = w * base; } r += 2 * m; } base = base * base; } if m > 1 { // m = 1 let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } } } } pub fn inv_fft(f: &mut [R], zeta_inv: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let zeta = zeta_inv; // inverse FFT let mut zetapow = Vec::with_capacity(20); { let mut m = 1; let mut cur = zeta; while m < n { zetapow.push(cur); cur = cur * cur; m *= 2; } } let mut m = 1; unsafe { if m < n { zetapow.pop(); let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } m = 2; } while m < n { let base = zetapow.pop().unwrap(); let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m) * w; *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = u - d; w = w * base; } r += 2 * m; } m *= 2; } } } } // Depends on: fft.rs, MInt.rs // Verified by: ABC269-Ex (https://atcoder.jp/contests/abc269/submissions/39116328) pub struct FPSOps { gen: mod_int::ModInt, } impl FPSOps { pub fn new(gen: mod_int::ModInt) -> Self { FPSOps { gen: gen } } } impl FPSOps { pub fn add(&self, mut a: Vec>, mut b: Vec>) -> Vec> { if a.len() < b.len() { std::mem::swap(&mut a, &mut b); } for i in 0..b.len() { a[i] += b[i]; } a } pub fn mul(&self, a: Vec>, b: Vec>) -> Vec> { type MInt = mod_int::ModInt; let n = a.len() - 1; let m = b.len() - 1; let mut p = 1; while p <= n + m { p *= 2; } let mut f = vec![MInt::new(0); p]; let mut g = vec![MInt::new(0); p]; for i in 0..n + 1 { f[i] = a[i]; } for i in 0..m + 1 { g[i] = b[i]; } let fac = MInt::new(p as i64).inv(); let zeta = self.gen.pow((M::m() - 1) / p as i64); fft::fft(&mut f, zeta, 1.into()); fft::fft(&mut g, zeta, 1.into()); for i in 0..p { f[i] *= g[i] * fac; } fft::inv_fft(&mut f, zeta.inv(), 1.into()); f.truncate(n + m + 1); f } } // Computes f^{-1} mod x^{f.len()}. // Reference: https://codeforces.com/blog/entry/56422 // Complexity: O(n log n) // Verified by: https://judge.yosupo.jp/submission/3219 // Depends on: MInt.rs, fft.rs fn fps_inv( f: &[mod_int::ModInt

], gen: mod_int::ModInt

) -> Vec> { let n = f.len(); assert!(n.is_power_of_two()); assert_eq!(f[0], 1.into()); let mut sz = 1; let mut r = vec![mod_int::ModInt::new(0); n]; let mut tmp_f = vec![mod_int::ModInt::new(0); n]; let mut tmp_r = vec![mod_int::ModInt::new(0); n]; r[0] = 1.into(); // Adopts the technique used in https://judge.yosupo.jp/submission/3153 while sz < n { let zeta = gen.pow((P::m() - 1) / sz as i64 / 2); tmp_f[..2 * sz].copy_from_slice(&f[..2 * sz]); tmp_r[..2 * sz].copy_from_slice(&r[..2 * sz]); fft::fft(&mut tmp_r[..2 * sz], zeta, 1.into()); fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into()); let fac = mod_int::ModInt::new(2 * sz as i64).inv().pow(2); for i in 0..2 * sz { tmp_f[i] *= tmp_r[i]; } fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into()); for v in &mut tmp_f[..sz] { *v = 0.into(); } fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into()); for i in 0..2 * sz { tmp_f[i] = -tmp_f[i] * tmp_r[i] * fac; } fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into()); r[sz..2 * sz].copy_from_slice(&tmp_f[sz..2 * sz]); sz *= 2; } r } type M = MInt; // Copied and modified from https://judge.yosupo.jp/submission/133199. // Originally by sansen. fn middle_product(c: &[M], a: &[M]) -> Vec { assert!(c.len() >= a.len()); if a.len() <= (1 << 5) { return c .windows(a.len()) .map(|c| { c.iter() .zip(a.iter()) .fold(MInt::new(0), |s, a| s + *a.0 * *a.1) }) .collect(); } let size = c.len().next_power_of_two(); let mut x = Vec::from(c); x.resize(size, MInt::new(0)); let mut y = Vec::from(a); y.reverse(); y.resize(size, MInt::new(0)); let zeta = MInt::new(3).pow((MOD - 1) / size as i64); fft::fft(&mut x, zeta, 1.into()); fft::fft(&mut y, zeta, 1.into()); let factor = MInt::new(size as i64).inv(); for i in 0..size { x[i] *= y[i] * factor; } fft::inv_fft(&mut x, zeta.inv(), 1.into()); (a.len()..=c.len()).map(|z| x[z - 1]).collect() } fn multipoint_evaluation(ops: &FPSOps, c: &[MInt], p: &[MInt]) -> Vec { if p.is_empty() { return vec![]; } let n = c.len(); let m = p.len(); let mut prod = vec![vec![]; 2 * m]; for (prod, p) in prod[m..].iter_mut().zip(p.iter()) { *prod = vec![MInt::new(1), -*p]; } for i in (1..m).rev() { prod[i] = ops.mul(prod[2 * i].clone(), prod[2 * i + 1].clone()); } let mut prod1 = prod[1].clone(); let mut sz = 1; while sz < n { sz *= 2; } prod1.resize(sz, 0.into()); let mut inv = fps_inv(&prod1, 3.into()); inv.truncate(n); let mut c = c.to_vec(); c.resize(n + m - 1, MInt::new(0)); let mut dp = vec![vec![]; 2 * m]; dp[1] = middle_product(&c, &inv); for i in 1..m { dp[2 * i] = middle_product(&dp[i], &prod[2 * i + 1]); dp[2 * i + 1] = middle_product(&dp[i], &prod[2 * i]); } dp[m..].iter().map(|dp| dp[0]).collect() } // End of copy-pasted part. fn fps_mul_all(ops: &FPSOps, f: &[Vec]) -> Vec { let m = f.len(); let mut seg = vec![vec![]; 2 * m]; for i in 0..m { seg[i + m] = f[i].to_vec(); } for i in (1..m).rev() { seg[i] = ops.mul( std::mem::replace(&mut seg[2 * i], vec![]), std::mem::replace(&mut seg[2 * i + 1], vec![]), ); } std::mem::replace(&mut seg[1], vec![]) } fn fps_common_denom(ops: &FPSOps, frac: &[(Vec, Vec)]) -> (Vec, Vec) { let m = frac.len(); let mut seg = vec![(vec![], vec![]); 2 * m]; for i in 0..m { seg[i + m] = frac[i].clone(); } for i in (1..m).rev() { let den = ops.mul(seg[2 * i].1.clone(), seg[2 * i + 1].1.clone()); let mut num = ops.mul( std::mem::replace(&mut seg[2 * i].1, vec![]), std::mem::replace(&mut seg[2 * i + 1].0, vec![]), ); let tmp = ops.mul( std::mem::replace(&mut seg[2 * i].0, vec![]), std::mem::replace(&mut seg[2 * i + 1].1, vec![]), ); num = ops.add(num, tmp); seg[i] = (num, den); } std::mem::replace(&mut seg[1], (vec![], vec![])) } // https://37zigen.com/lagrange-interpolation/ fn lagrange_interpolate(ops: &FPSOps, xy: &[(MInt, MInt)]) -> Vec { let n = xy.len(); let mut xs = vec![MInt::new(0); n]; let mut ps = vec![vec![]; n]; for i in 0..n { xs[i] = xy[i].0; ps[i] = vec![-xy[i].0, 1.into()]; } let g = fps_mul_all(ops, &ps); let mut gdash = vec![MInt::new(0); n]; for i in 0..n { gdash[i] = g[i + 1] * (i + 1) as i64; } let vals = multipoint_evaluation(ops, &gdash, &xs); let mut fracs = vec![(vec![MInt::new(1)], vec![]); n]; for i in 0..n { fracs[i].0[0] = vals[i].inv() * xy[i].1; fracs[i].1 = vec![-xy[i].0, 1.into()]; } let (num, _) = fps_common_denom(ops, &fracs); num } // https://yukicoder.me/problems/no/1938 (4) // X がいずれかの x_i と等しい場合、F(X) = (N-1)y_i + F_i(x_i) だから 1 回多項式補間をするだけでよい。 // そうでない場合、G(x) を (x_i, y_i) すべてで補間した N-1 次多項式とすると、 // F(X) = NG(X) - (\sum_i A_i / (X - x_i)) (\prod (X - x_i)) where A_i := (y_i - F_i(x_i)) / \prod_{j != i} (x_i - x_j) である。 // A_i はすべて [x^{N-1}]G(x) に等しいため、これは計算できる。 // 計算量は多項式補間がボトルネックであるため O(N log^2 N) である。 fn main() { input! { n: usize, bigx: i64, xy: [(i64, i64); n], } let ops = FPSOps { gen: 3.into(), }; let xy: Vec<_> = xy.into_iter().map(|(x, y)| (MInt::new(x), MInt::new(y))).collect(); let mut idx = n; for i in 0..n { if MInt::new(bigx) == xy[i].0 { idx = i; break; } } if idx < n { let mut rm = xy.clone(); rm.remove(idx); let p = lagrange_interpolate(&ops, &rm); let mut ans = xy[idx].1 * (n - 1) as i64; let mut cur = MInt::new(1); for i in 0..p.len() { ans += cur * p[i]; cur *= xy[idx].0; } println!("{}", ans); return; } let g = lagrange_interpolate(&ops, &xy); let lead = g[n - 1]; let mut fracs = vec![(vec![MInt::new(1)], vec![]); n]; for i in 0..n { fracs[i].1 = vec![-xy[i].0, 1.into()]; } let (num, _) = fps_common_denom(&ops, &fracs); let mut ans = MInt::new(0); let mut cur = MInt::new(1); for i in 0..n { ans += cur * (g[i] * n as i64 - num[i] * lead); cur *= bigx; } println!("{}", ans); }