結果
| 問題 | No.1962 Not Divide |
| ユーザー |
 uwi
|
| 提出日時 | 2022-06-09 18:26:41 |
| 言語 | Java21 (openjdk 21) |
| 結果 |
AC
|
| 実行時間 | 802 ms / 2,000 ms |
| コード長 | 32,567 bytes |
| コンパイル時間 | 6,608 ms |
| コンパイル使用メモリ | 98,800 KB |
| 実行使用メモリ | 62,560 KB |
| 最終ジャッジ日時 | 2024-09-21 05:37:43 |
| 合計ジャッジ時間 | 19,843 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge3 |
（要ログイン）
テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 246 ms
57,628 KB |
| testcase_01 | AC | 245 ms
57,328 KB |
| testcase_02 | AC | 404 ms
59,896 KB |
| testcase_03 | AC | 686 ms
62,560 KB |
| testcase_04 | AC | 339 ms
61,500 KB |
| testcase_05 | AC | 655 ms
60,280 KB |
| testcase_06 | AC | 744 ms
62,356 KB |
| testcase_07 | AC | 393 ms
59,464 KB |
| testcase_08 | AC | 710 ms
62,500 KB |
| testcase_09 | AC | 568 ms
62,536 KB |
| testcase_10 | AC | 420 ms
60,600 KB |
| testcase_11 | AC | 609 ms
60,116 KB |
| testcase_12 | AC | 467 ms
60,376 KB |
| testcase_13 | AC | 419 ms
60,344 KB |
| testcase_14 | AC | 667 ms
62,472 KB |
| testcase_15 | AC | 364 ms
60,200 KB |
| testcase_16 | AC | 399 ms
59,240 KB |
| testcase_17 | AC | 639 ms
62,028 KB |
| testcase_18 | AC | 247 ms
58,284 KB |
| testcase_19 | AC | 223 ms
56,308 KB |
| testcase_20 | AC | 793 ms
62,540 KB |
| testcase_21 | AC | 802 ms
62,360 KB |
| testcase_22 | AC | 773 ms
60,148 KB |
| testcase_23 | AC | 760 ms
62,188 KB |
ソースコード
package contest220527;
import java.io.*;
import java.util.ArrayDeque;
import java.util.Arrays;
import java.util.InputMismatchException;
import java.util.Queue;
import java.util.function.IntUnaryOperator;
import java.util.function.LongUnaryOperator;
public class FX {
InputStream is;
FastWriter out;
String INPUT = "";
public void solve()
{
// 数字iが連続する遷移を、
// f_i = sum_{j:iの倍数でない} x^j
// で表す。
// また、本来求めたい多項式をHとし、数字iで終わる個数の多項式をh_iとすると、
// H = sum_i h_iである。ただし長さ0のところは0とする。
// 現在の数以外の遷移なので、次の式が成り立つ。
// h_i = (H-h_i+1)f_i
// h_i(1+f_i) = (H+1)f_i
// H = sum_i (H+1)f_i/(1+f_i)
// H/(H+1) = sum_i f_i/(1+f_i)
// Hが何次式かはわからないけど、右辺は計算できる
// f_i = 1/(1-x) - 1/(1-x^i)
int n = ni(), m = ni();
// 1-1/(1+f_i)
int D = 11000;
long[] r = new long[D];
r[0] = m-1;
for(int i = 2;i <= m;i++){
// (x-1)(x^i-1)/((x-2)x^i+1)
long[] f = new long[D];
f[0] = 1;
f[1] = mod - 1;
f[i+1] = 1;
f[i] = mod - 1;
for(int j = 2;j < D;j++){
if(j-i >= 0)f[j] += f[j-i] * 2;
if(j-i-1 >= 0)f[j] -= f[j-i-1];
f[j] %= mod;
if(f[j] < 0)f[j] += mod;
}
r = sub(r, f);
}
// tr(r);
// H/(H+1) = r
// H = (H+1)r
// H = 1/(1-r)-1
long[] den = sub(new long[]{1}, r);
long[] H = sub(inv(den), new long[]{1});
H[0] = 1;
out.println(guess(H, mod, n));
}
public static long bostanMori(long[] P, long[] Q, long n)
{
// P(x)Q(-x)
while(n > 0) {
if(Q.length > n+1)Q = Arrays.copyOf(Q, (int)n+1);
if(P.length > n+1)P = Arrays.copyOf(P, (int)n+1);
long[] QF = Arrays.copyOf(Q, Q.length);
for (int i = 1; i < QF.length; i += 2) {
QF[i] = Q[i] == 0 ? 0 : mod - Q[i];
}
long[] PQF = mul(P, QF);
long[] QQF = mul(Q, QF);
Q = even(QQF);
P = n % 2 == 0 ? even(PQF) : odd(PQF);
n /= 2;
}
return P[0] * invl(Q[0], mod) % mod;
}
private static long[] even(long[] a)
{
long[] ret = new long[(a.length+1)/2];
for(int i = 0;i < a.length;i+=2){
ret[i/2] = a[i];
}
return ret;
}
private static long[] odd(long[] a)
{
long[] ret = new long[a.length/2];
for(int i = 1;i < a.length;i+=2){
ret[i/2] = a[i];
}
return ret;
}
public long guess(long[] a, int mod, long n)
{
if(a.length % 2 == 1)a = Arrays.copyOf(a, a.length/2*2); // strip to even length
long[] q = modifiedBerlekampMassey(a, mod);
q = rev(q);
long[] aa = mul(a, q, q.length-1);
return bostanMori(aa, q, n);
}
public static long[] rev(long[] a){long[] b = new long[a.length];for(int i = 0;i < a.length;i++)b[a.length-1-i] = a[i];return b;}
public static long lr(long[] a, long[] co, long n, long mod)
{
int m = a.length;
if(m == 1){
long ret = a[0];
long mul = co[0];
for(;n > 0;n >>>= 1){
if((n&1)==1){
ret = (ret * mul) % mod;
}
mul = (mul * mul) % mod;
}
return ret;
}
long[][] m2 = new long[m-1][m];
System.arraycopy(co, 0, m2[0], 0, m);
for(int i = 1;i < m-1;i++){
for(int j = 0;j < m;j++){
long x = m2[i-1][m-1] * co[j];
if(j-1 >= 0)x += m2[i-1][j-1];
m2[i][j] = x % mod;
}
}
long[] ret = new long[m];
ret[0] = 1;
long[] temp = new long[2*m];
for(int l = 62;l >= 0;l--){
if(n<<~l<0){
for(int i = 0;i < m;i++)temp[i] = ret[m-1] * co[i];
for(int i = 1;i < m;i++)temp[i] += ret[i-1];
for(int i = 0;i < m;i++)ret[i] = temp[i] % mod;
}
if(l > 0){
Arrays.fill(temp, 0L);
for(int i = 0;i < m;i++){
for(int j = 0;j < m;j++){
temp[i+j] += ret[i] * ret[j] % mod;
}
}
for(int i = 0;i < 2*m-1;i++){
temp[i] %= mod;
}
for(int i = 0;i < m;i++){
long s = temp[i];
for(int j = m;j < 2*m-1;j++){
s += temp[j] * m2[j-m][i] % mod;
}
ret[i] = s % mod;
}
}
}
long s = 0;
for(int i = 0;i < m;i++){
s += ret[i] * a[i] % mod;
}
return s % mod;
}
public static long[] modifiedBerlekampMassey(long[] a, int mod)
{
assert a.length % 2 == 0;
int n = a.length/2;
int m = 2*n-1;
long[] R0 = new long[2*n+1]; R0[2*n] = 1;
long[] R1 = new long[2*n];
for(int i = 0;i < 2*n;i++)R1[i] = a[m-i];
R1 = strip(R1);
long[][] IM0 = hgcd(R0, R1);
long[] v1 = IM0[3];
// long[] v0 = {0L};
// long[] v1 = {1L};
// while(n <= R1.length-1){
// long[][] QR = divmod(R0, R1, mod);
// long[] v = sub(v0, mul(QR[0], v1));
// v0 = v1; v1 = v; R0 = R1; R1 = QR[1];
// }
// tr2(R0);
// tr2(R1);
// tr2(v1);
// tr2("high");
// tr2(xv1);
// for(long[] u : res){
// tr2(u);
// }
// normalize
long z = invl(v1[v1.length-1], mod);
for(int i = 0;i < v1.length;i++){
v1[i] = v1[i] * z % mod;
}
return strip(v1);
}
private static long[][] cogcdOnce(long[] p0, long[] p1)
{
assert p0.length > p1.length;
long[][] ret = E;
long[][] IM0 = hgcd(p0, p1);
long[] p3 = mul2low(IM0, p0, p1);
ret = mul22(IM0, ret);
if (p3.length == 0) return ret;
// long[] p2 = mul2high(IM0, p0, p1);
// long[][] qr = div(p2, p3);
// long[] p4 = qr[1];
// ret = mul22q(qr[0], ret);
// if (p4.length == 0) return ret;
return ret;
}
private static void tr2(Object... o) { System.out.println(Arrays.deepToString(o)); }
public static long[] strip(long[] a)
{
int i;
for(i = a.length-1;i > 0 && a[i] == 0;i--);
if(i + 1 == a.length)return a;
return Arrays.copyOf(a, i+1);
}
public static long[][] divmod(long[] A, long[] B, int mod)
{
assert B[B.length-1] != 0;
long[] R = Arrays.copyOf(A, A.length);
long[] Q = new long[Math.max(1, A.length-B.length+1)];
long ib = invl(B[B.length-1], mod);
for(int i = A.length-1, t = Q.length-1;i >= B.length-1;i--,t--){
long m = R[i] * ib % mod;
Q[t] = m;
for(int j = 0, k = i-B.length+1+j;j < B.length;j++,k++){
R[k] -= B[j] * m;
R[k] %= mod;
if(R[k] < 0)R[k] += mod;
}
assert R[i] == 0;
}
return new long[][]{Q, strip(R)};
}
private static final long[][] E = {{1},{},{},{1}};
public static long[] gcd(long[] a, long[] b)
{
a = clean(a); b = clean(b);
if(a.length < b.length){
long[] d = a; a = b; b = d;
}
if (a.length == b.length) {
long t = ((long)mod*mod - invl(a[a.length-1], mod) * b[b.length-1]) % mod;
for(int i = 0;i < b.length;i++){
b[i] = (b[i] + a[i] * t) % mod;
}
b = clean(b);
assert a.length > b.length;
}
long[][] ico = cogcd(a, b);
long[] ret = clean(mul2high(ico, a, b));
// normalize (to monic)
long c = invl(ret[ret.length-1], mod);
for(int i = 0;i < ret.length;i++)ret[i] = ret[i] * c % mod;
return ret;
}
public static long[][] exgcd(long[] a, long[] b)
{
a = clean(a); b = clean(b);
if(a.length < b.length){
long[][] ret = exgcd(b, a);
return new long[][]{ret[1], ret[0], ret[2]};
}
if (a.length == b.length) {
long t = ((long)mod*mod - invl(a[a.length-1], mod) * b[b.length-1]) % mod;
for(int i = 0;i < b.length;i++){
b[i] = (b[i] + a[i] * t) % mod;
}
b = clean(b);
assert a.length > b.length;
long[][] ret = exgcd(a, b);
// (A B)(a ) = (g)
// (C D)(b+ta) (0)
// (A+tB B)(a) = (g)
// (C+tD D)(b) (0)
ret[0] = clean(add(ret[0], mul(ret[1], t)));
return ret;
}
long[][] ico = cogcd(a, b);
long[] gcd = clean(mul2high(ico, a, b));
return new long[][]{ico[0], ico[1], gcd};
}
private static long[][] cogcd(long[] p0, long[] p1)
{
assert p0.length > p1.length;
long[][] ret = E;
while(true) {
long[][] IM0 = hgcd(p0, p1);
long[] p3 = mul2low(IM0, p0, p1);
ret = mul22(IM0, ret);
if (p3.length == 0) return ret;
long[] p2 = mul2high(IM0, p0, p1);
long[][] qr = div(p2, p3);
long[] p4 = qr[1];
ret = mul22q(qr[0], ret);
if (p4.length == 0) return ret;
p0 = p3; p1 = p4;
}
}
private static long[][] hgcd(long[] a, long[] b)
{
if(b.length == 0)return E;
int N = a.length-1;
int M = b.length-1;
assert N > M;
int m = (N+1)/2; // the magic threshold
if(M < m)return E;
long[] a0 = Arrays.copyOfRange(a, m, a.length);
long[] b0 = Arrays.copyOfRange(b, m, b.length);
long[][] IR = hgcd(a0, b0);
long[] bp = mul2low(IR, a, b);
if(bp.length-1 < m)return IR;
long[] ap = mul2high(IR, a, b);
long[][] qr = div(ap, bp);
long[] D = qr[1], C = bp;
int l = bp.length-1;
int k = 2*m-l;
long[] C0 = Arrays.copyOfRange(C, k, C.length);
long[] D0 = Arrays.copyOfRange(D, k, D.length);
long[][] IS = hgcd(C0, D0);
return clean(mul22(mul22q(IS, qr[0]), IR));
}
/**
* P(x)^nをm次まで求める。
*
* Q(x)=P(x)^nとすると、
* Q'(x)=nP'(x)P(x)^{n-1}である。したがって、
* Q(x) = P(x) * Q'(x)/n/P'(x)
* nP'(x)Q(x) = P(x)Q'(x)である。
* これのx^iの係数は、
* n(sum_j (i-j+1)p[i-j+1]*q[j]) = sum_j p[i-j]*(j+1)q[j+1]
* となる。
* ここから、
* q[i+1] = (n(sum_j (i-j+1)p[i-j+1]*q[j]) - sum_{j=0}^{i-1} p[i-j]*(j+1)q[j+1]) / p[0] / (i+1)
* が導かれる。sumは、iが大きくなっても|P|で抑えられるので、全体でO(|P|m)になる。
* 0<=i-j+1<|P| -> i+1-|P|<j<=i+1
*
* またこれはnが負のときでも成立する。
*
* @param P P[0] != 0
* @param n
* @param m
* @return
*/
public static long[] pow(long[] P, int n, int m)
{
long[] Q = new long[m+1];
long ip0 = invl(P[0], mod);
Q[0] = n >= 0 ? pow(P[0], n, mod) : pow(ip0, n, mod);
for(int i = 0;i < m;i++){
long s = 0;
for(int j = Math.max(0, i+1-P.length+1);j <= i;j++){
s += (i-j+1) * P[i-j+1] % mod * Q[j];
if(s >= big)s -= big;
}
s %= mod;
long t = 0;
for(int j = Math.max(0, i-P.length+1);j <= i-1;j++){
t += (j+1) * P[i-j] % mod * Q[j+1];
if(t >= big)t -= big;
}
t %= mod;
s = (s*n-t) % mod;
if(s < 0)s += mod;
Q[i+1] = s * ip0 % mod * invl(i+1, mod) % mod;
}
return Q;
}
/**
* Pがsparseな場合のP^nをm次まで
* O(|P|m).
* NOT VERIFIED
*
* @param P [index, value] P[0] != 0
* @param n
* @param m
* @return
*/
public static long[] pow(long[][] P, int n, int m)
{
long[] Q = new long[m+1];
long p0 = 0;
for(long[] u : P)if(u[0] == 0)p0 = u[1];
assert p0 != 0;
long ip0 = invl(p0, mod);
Q[0] = n >= 0 ? pow(p0, n, mod) : pow(ip0, n, mod);
for(int i = 0;i < m;i++){
long s = 0;
for (long[] u : P) {
if (Math.max(0, i + 1 - P.length + 1) <= i - u[0] + 1 && i - u[0] + 1 <= i) {
s += u[0] * u[1] % mod * Q[i - (int) u[0] + 1];
if(s >= big)s -= big;
}
}
s %= mod;
long t = 0;
for(long[] u : P) {
if (Math.max(0, i - P.length + 1) <= i - u[0] && i - u[0] <= i - 1) {
t += (i-u[0]+1) * u[1] % mod * Q[i - (int) u[0] + 1];
if(t >= big)t -= big;
}
}
t %= mod;
s = (s*n-t) % mod;
if(s < 0)s += mod;
Q[i+1] = s * ip0 % mod * invl(i+1, mod) % mod;
}
return Q;
}
/**
* n=500000, K=10^9でpowより1.76倍遅い
* @param a
* @param K
* @return
*/
public static long[] powNaive(long[] a, int K)
{
int n = a.length;
long[] ret = {1};
for(int d = 31-Integer.numberOfLeadingZeros(K);d >= 0;d--) {
ret = mul(ret, ret, n);
if(K<<~d<0) {
ret = mul(ret, a, n);
}
}
return ret;
}
public static long pow(long a, long n, long mod) {
// a %= mod;
long ret = 1;
int x = 63 - Long.numberOfLeadingZeros(n);
for (; x >= 0; x--) {
ret = ret * ret % mod;
if (n << 63 - x < 0)
ret = ret * a % mod;
}
return ret;
}
public static long[] reverse_(long[] p)
{
for(int i = 0, j = p.length-1;i < j;i++,j--){
long d = p[i]; p[i] = p[j]; p[j] = d;
}
return p;
}
public static long[] reverse(long[] p)
{
long[] ret = new long[p.length];
for(int i = 0;i < p.length;i++){
ret[i] = p[p.length-1-i];
}
return ret;
}
public static long[] reverse(long[] p, int lim)
{
long[] ret = new long[lim];
for(int i = 0;i < lim && i < p.length;i++){
ret[i] = p[p.length-1-i];
}
return ret;
}
// [quotient, remainder]
// remainder can be empty.
//
// deg(f)=n, deg(g)=m, f=gq+r, f=gq+r.
// f* = x^n*f(1/x),
// t=g*^-1 mod x^(n-m+1), q=(tf* mod x^(n-m+1))*
public static long[][] div(long[] f, long[] g)
{
int n = f.length, m = g.length;
if(n < m)return new long[][]{new long[0], Arrays.copyOf(f, n)};
long[] rf = reverse(f, n-m+1);
long[] rg = reverse(g, n-m+1);
long[] rq = mul(rf, inv(rg), n-m+1);
long[] q = reverse(rq, n-m+1);
long[] r = sub(f, mul(q, g, m-1), m-1);
return new long[][]{q, r};
}
static long[] mul2high(long[][] x, long[] a, long[] b)
{
return clean(add(mul(x[0], a), mul(x[1], b)));
}
static long[] mul2low(long[][] x, long[] a, long[] b)
{
return clean(add(mul(x[2], a), mul(x[3], b)));
}
static long[] clean(long[] a)
{
for(int i = a.length-1;i >= 0;i--){
if(a[i] != 0)return i == a.length-1 ? a : Arrays.copyOf(a, i+1);
}
return new long[0];
}
static long[][] clean(long[][] a)
{
for(int i = 0;i < a.length;i++)a[i] = clean(a[i]);
return a;
}
static long[][] mul22(long[][] A, long[][] B)
{
assert A.length == 4;
assert B.length == 4;
long[][] C = new long[4][];
C[0] = clean(add(mul(A[0], B[0]), mul(A[1], B[2])));
C[1] = clean(add(mul(A[0], B[1]), mul(A[1], B[3])));
C[2] = clean(add(mul(A[2], B[0]), mul(A[3], B[2])));
C[3] = clean(add(mul(A[2], B[1]), mul(A[3], B[3])));
return C;
}
static long[][] mul22q(long[][] A, long[] q)
{
assert A.length == 4;
long[][] C = new long[4][];
C[0] = flip(A[1]);
C[1] = clean(sub(mul(A[1], q), A[0]));
C[2] = flip(A[3]);
C[3] = clean(sub(mul(A[3], q), A[2]));
return C;
}
static long[][] mul22q(long[] q, long[][] A)
{
assert A.length == 4;
long[][] C = new long[4][];
C[0] = flip(A[2]);
C[1] = flip(A[3]);
C[2] = clean(sub(mul(A[2], q), A[0]));
C[3] = clean(sub(mul(A[3], q), A[1]));
return C;
}
static long[] flip(long[] a)
{
long[] ret = Arrays.copyOf(a, a.length);
for(int i = 0;i < a.length;i++){
ret[i] = mod - a[i];
if(ret[i] == mod)ret[i] = 0;
}
return ret;
}
public static final int mod = 998244353;
public static final int G = 3;
// only 998244353
public static long[] mul(long[] a, long[] b)
{
if(a.length == 0 && b.length == 0)return new long[0];
if(a.length + b.length >= 300) {
return Arrays.copyOf(NTTStockham998244353.convolve(a, b), a.length + b.length - 1);
}else{
return mulnaive(a, b);
}
}
public static long[] mul(long[] a, long[] b, int lim)
{
if(a.length + b.length >= 300) {
return Arrays.copyOf(NTTStockham998244353.convolve(a, b), lim);
}else{
return mulnaive(a, b, lim);
}
}
// public static final int mod = 1000000007;
// public static long[] mul(long[] a, long[] b)
// {
// if(Math.max(a.length, b.length) >= 3000){
// return Arrays.copyOf(NTTCRT.convolve(a, b, 3, mod), a.length+b.length-1);
// }else{
// return mulnaive(a, b);
// }
// }
// public static long[] mul(long[] a, long[] b, int lim)
// {
// if(Math.max(a.length, b.length) >= 3000){
// return Arrays.copyOf(NTTCRT.convolve(a, b, 3, mod), lim);
// }else{
// return mulnaive(a, b, lim);
// }
// }
public static final long big = (Long.MAX_VALUE/mod/mod-1)*mod*mod;
public static long[] mulnaive(long[] a, long[] b)
{
long[] c = new long[a.length+b.length-1];
for(int i = 0;i < a.length;i++){
for(int j = 0;j < b.length;j++){
c[i+j] += a[i]*b[j];
if(c[i+j] >= big)c[i+j] -= big;
}
}
for(int i = 0;i < c.length;i++)c[i] %= mod;
return c;
}
public static long[] mulnaive(long[] a, long[] b, int lim)
{
long[] c = new long[lim];
for(int i = 0;i < a.length;i++){
for(int j = 0;j < b.length && i+j < lim;j++){
c[i+j] += a[i]*b[j];
if(c[i+j] >= big)c[i+j] -= big;
}
}
for(int i = 0;i < c.length;i++)c[i] %= mod;
return c;
}
public static long[] mul_(long[] a, long k)
{
for(int i = 0;i < a.length;i++)a[i] = a[i] * k % mod;
return a;
}
public static long[] mul(long[] a, long k)
{
a = Arrays.copyOf(a, a.length);
for(int i = 0;i < a.length;i++)a[i] = a[i] * k % mod;
return a;
}
public static long[] add(long[] a, long[] b)
{
long[] c = new long[Math.max(a.length, b.length)];
for(int i = 0;i < a.length;i++)c[i] += a[i];
for(int i = 0;i < b.length;i++)c[i] += b[i];
for(int i = 0;i < c.length;i++)if(c[i] >= mod)c[i] -= mod;
return c;
}
public static long[] add(long[] a, long[] b, int lim)
{
long[] c = new long[lim];
for(int i = 0;i < a.length && i < lim;i++)c[i] += a[i];
for(int i = 0;i < b.length && i < lim;i++)c[i] += b[i];
for(int i = 0;i < c.length;i++)if(c[i] >= mod)c[i] -= mod;
return c;
}
public static long[] sub(long[] a, long[] b)
{
long[] c = new long[Math.max(a.length, b.length)];
for(int i = 0;i < a.length;i++)c[i] += a[i];
for(int i = 0;i < b.length;i++)c[i] -= b[i];
for(int i = 0;i < c.length;i++)if(c[i] < 0)c[i] += mod;
return c;
}
public static long[] sub(long[] a, long[] b, int lim)
{
long[] c = new long[lim];
for(int i = 0;i < a.length && i < lim;i++)c[i] += a[i];
for(int i = 0;i < b.length && i < lim;i++)c[i] -= b[i];
for(int i = 0;i < c.length;i++)if(c[i] < 0)c[i] += mod;
return c;
}
// F_{t+1}(x) = -F_t(x)^2*P(x) + 2F_t(x)
// if want p-destructive, comment out flipping p just before returning.
public static long[] inv(long[] p)
{
int n = p.length;
long[] f = {invl(p[0], mod)};
for(int i = 0;i < p.length;i++){
if(p[i] == 0)continue;
p[i] = mod-p[i];
}
for(int i = 1;i < 2*n;i*=2){
long[] f2 = mul(f, f, Math.min(n, 2*i));
long[] f2p = mul(f2, Arrays.copyOf(p, i), Math.min(n, 2*i));
for(int j = 0;j < f.length;j++){
f2p[j] += 2L*f[j];
if(f2p[j] >= mod)f2p[j] -= mod;
if(f2p[j] >= mod)f2p[j] -= mod;
}
f = f2p;
}
for(int i = 0;i < p.length;i++){
if(p[i] == 0)continue;
p[i] = mod-p[i];
}
return f;
}
// differentiate
public static long[] d(long[] p)
{
long[] q = new long[p.length];
for(int i = 0;i < p.length-1;i++){
q[i] = p[i+1] * (i+1) % mod;
}
return q;
}
// integrate
public static long[] i(long[] p)
{
long[] q = new long[p.length];
for(int i = 0;i < p.length-1;i++){
q[i+1] = p[i] * invl(i+1, mod) % mod;
}
return q;
}
public static long invl(long a, long mod) {
long b = mod;
long p = 1, q = 0;
while (b > 0) {
long c = a / b;
long d;
d = a;
a = b;
b = d % b;
d = p;
p = q;
q = d - c * q;
}
return p < 0 ? p + mod : p;
}
public static class NTTStockham998244353 {
private static final int P = 998244353, mod = P, G = 3;
private static long[] wps;
public static long[] convolve(long[] a, long[] b)
{
int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
wps = new long[m];
long unit = pow(G, (P-1)/m);
wps[0] = 1;
for(int p = 1;p < m;p++) {
wps[p] = wps[p-1] * unit % mod;
}
long[] fa = go(a, m, false);
long[] fb = a == b ? fa : go(b, m, false);
for(int i = 0;i < m;i++){
fa[i] = fa[i]*fb[i] % mod;
}
fa = go(fa, m, true);
for(int i = 1, j = m-1;i < j;i++,j--) {
long d = fa[i]; fa[i] = fa[j]; fa[j] = d;
}
return fa;
}
private static void fft(long[] X, long[] Y)
{
int s = 1;
boolean eo = false;
for(int n = X.length;n >= 4;n /= 2) {
int m = n/2;
for(int p = 0;p < m;p++) {
long wp = wps[s*p];
long wk = (wp<<32)/P;
for(int q = 0;q < s;q++) {
long a = X[q + s*(p+0)];
long b = X[q + s*(p+m)];
long ndsts = a + b;
if(ndsts >= 2*P)ndsts -= 2*P;
long T = a - b + 2*P;
long Q = wk*T>>>32;
Y[q + s*(2*p+0)] = ndsts;
Y[q + s*(2*p+1)] = wp*T-Q*P&(1L<<32)-1;
}
}
s *= 2;
eo = !eo;
long[] D = X; X = Y; Y = D;
}
long[] z = eo ? Y : X;
for(int q = 0;q < s;q++) {
long a = X[q + 0];
long b = X[q + s];
z[q+0] = (a+b) % P;
z[q+s] = (a-b+2*P) % P;
}
}
// private static void fft(long[] X, long[] Y)
// {
// int s = 1;
// boolean eo = false;
// for(int n = X.length;n >= 4;n /= 2) {
// int m = n/2;
// for(int p = 0;p < m;p++) {
// long wp = wps[s*p];
// for(int q = 0;q < s;q++) {
// long a = X[q + s*(p+0)];
// long b = X[q + s*(p+m)];
// Y[q + s*(2*p+0)] = (a+b) % P;
// Y[q + s*(2*p+1)] = (a-b+P) * wp % P;
// }
// }
// s *= 2;
// eo = !eo;
// long[] D = X; X = Y; Y = D;
// }
// long[] z = eo ? Y : X;
// for(int q = 0;q < s;q++) {
// long a = X[q + 0];
// long b = X[q + s];
// z[q+0] = (a+b) % P;
// z[q+s] = (a-b+P) % P;
// }
// }
private static long[] go(long[] src, int n, boolean inverse)
{
long[] dst = Arrays.copyOf(src, n);
fft(dst, new long[n]);
if(inverse){
long in = invl(n);
for(int i = 0;i < n;i++){
dst[i] = dst[i] * in % mod;
}
}
return dst;
}
private static long pow(long a, long n) {
// a %= mod;
long ret = 1;
int x = 63 - Long.numberOfLeadingZeros(n);
for (; x >= 0; x--) {
ret = ret*ret % mod;
if (n<<~x<0)ret = ret*a%mod;
}
return ret;
}
private static long invl(long a) {
long b = mod;
long p = 1, q = 0;
while (b > 0) {
long c = a / b;
long d;
d = a;
a = b;
b = d % b;
d = p;
p = q;
q = d - c * q;
}
return p < 0 ? p + mod : p;
}
}
public static void main(String[] args) {
new FX().run();
}
public void run()
{
long S = System.currentTimeMillis();
is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
out = new FastWriter(System.out);
solve();
out.flush();
long G = System.currentTimeMillis();
tr(G-S+"ms");
// Thread t = new Thread(null, null, "~", Runtime.getRuntime().maxMemory()){
// @Override
// public void run() {
// long s = System.currentTimeMillis();
// solve();
// out.flush();
// if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
// }
// };
// t.start();
// t.join();
}
private boolean eof()
{
if(lenbuf == -1)return true;
int lptr = ptrbuf;
while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
try {
is.mark(1000);
while(true){
int b = is.read();
if(b == -1){
is.reset();
return true;
}else if(!isSpaceChar(b)){
is.reset();
return false;
}
}
} catch (IOException e) {
return true;
}
}
private final byte[] inbuf = new byte[1024];
public int lenbuf = 0, ptrbuf = 0;
private int readByte()
{
if(lenbuf == -1)throw new InputMismatchException();
if(ptrbuf >= lenbuf){
ptrbuf = 0;
try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
if(lenbuf <= 0)return -1;
}
return inbuf[ptrbuf++];
}
private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
// private boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
private double nd() { return Double.parseDouble(ns()); }
private char nc() { return (char)skip(); }
private String ns()
{
int b = skip();
StringBuilder sb = new StringBuilder();
while(!(isSpaceChar(b))){
sb.appendCodePoint(b);
b = readByte();
}
return sb.toString();
}
private char[] ns(int n)
{
char[] buf = new char[n];
int b = skip(), p = 0;
while(p < n && !(isSpaceChar(b))){
buf[p++] = (char)b;
b = readByte();
}
return n == p ? buf : Arrays.copyOf(buf, p);
}
private char[][] nm(int n, int m)
{
char[][] map = new char[n][];
for(int i = 0;i < n;i++)map[i] = ns(m);
return map;
}
private int[][] nmi(int n, int m)
{
int[][] map = new int[n][];
for(int i = 0;i < n;i++)map[i] = na(m);
return map;
}
private int[] na(int n)
{
int[] a = new int[n];
for(int i = 0;i < n;i++)a[i] = ni();
return a;
}
private long[] nal(int n)
{
long[] a = new long[n];
for(int i = 0;i < n;i++)a[i] = nl();
return a;
}
private int ni()
{
int num = 0, b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
b = readByte();
}
while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
b = readByte();
}
}
private long nl()
{
long num = 0;
int b;
boolean minus = false;
while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
if(b == '-'){
minus = true;
b = readByte();
}
while(true){
if(b >= '0' && b <= '9'){
num = num * 10 + (b - '0');
}else{
return minus ? -num : num;
}
b = readByte();
}
}
public static class FastWriter
{
private static final int BUF_SIZE = 1<<13;
private final byte[] buf = new byte[BUF_SIZE];
private final OutputStream out;
private int ptr = 0;
private FastWriter(){out = null;}
public FastWriter(OutputStream os)
{
this.out = os;
}
public FastWriter(String path)
{
try {
this.out = new FileOutputStream(path);
} catch (FileNotFoundException e) {
throw new RuntimeException("FastWriter");
}
}
public FastWriter write(byte b)
{
buf[ptr++] = b;
if(ptr == BUF_SIZE)innerflush();
return this;
}
public FastWriter write(char c)
{
return write((byte)c);
}
public FastWriter write(char[] s)
{
for(char c : s){
buf[ptr++] = (byte)c;
if(ptr == BUF_SIZE)innerflush();
}
return this;
}
public FastWriter write(String s)
{
s.chars().forEach(c -> {
buf[ptr++] = (byte)c;
if(ptr == BUF_SIZE)innerflush();
});
return this;
}
private static int countDigits(int l) {
if (l >= 1000000000) return 10;
if (l >= 100000000) return 9;
if (l >= 10000000) return 8;
if (l >= 1000000) return 7;
if (l >= 100000) return 6;
if (l >= 10000) return 5;
if (l >= 1000) return 4;
if (l >= 100) return 3;
if (l >= 10) return 2;
return 1;
}
public FastWriter write(int x)
{
if(x == Integer.MIN_VALUE){
return write((long)x);
}
if(ptr + 12 >= BUF_SIZE)innerflush();
if(x < 0){
write((byte)'-');
x = -x;
}
int d = countDigits(x);
for(int i = ptr + d - 1;i >= ptr;i--){
buf[i] = (byte)('0'+x%10);
x /= 10;
}
ptr += d;
return this;
}
private static int countDigits(long l) {
if (l >= 1000000000000000000L) return 19;
if (l >= 100000000000000000L) return 18;
if (l >= 10000000000000000L) return 17;
if (l >= 1000000000000000L) return 16;
if (l >= 100000000000000L) return 15;
if (l >= 10000000000000L) return 14;
if (l >= 1000000000000L) return 13;
if (l >= 100000000000L) return 12;
if (l >= 10000000000L) return 11;
if (l >= 1000000000L) return 10;
if (l >= 100000000L) return 9;
if (l >= 10000000L) return 8;
if (l >= 1000000L) return 7;
if (l >= 100000L) return 6;
if (l >= 10000L) return 5;
if (l >= 1000L) return 4;
if (l >= 100L) return 3;
if (l >= 10L) return 2;
return 1;
}
public FastWriter write(long x)
{
if(x == Long.MIN_VALUE){
return write("" + x);
}
if(ptr + 21 >= BUF_SIZE)innerflush();
if(x < 0){
write((byte)'-');
x = -x;
}
int d = countDigits(x);
for(int i = ptr + d - 1;i >= ptr;i--){
buf[i] = (byte)('0'+x%10);
x /= 10;
}
ptr += d;
return this;
}
public FastWriter write(double x, int precision)
{
if(x < 0){
write('-');
x = -x;
}
x += Math.pow(10, -precision)/2;
// if(x < 0){ x = 0; }
write((long)x).write(".");
x -= (long)x;
for(int i = 0;i < precision;i++){
x *= 10;
write((char)('0'+(int)x));
x -= (int)x;
}
return this;
}
public FastWriter writeln(char c){ return write(c).writeln(); }
public FastWriter writeln(int x){ return write(x).writeln(); }
public FastWriter writeln(long x){ return write(x).writeln(); }
public FastWriter writeln(double x, int precision){ return write(x, precision).writeln(); }
public FastWriter write(int... xs)
{
boolean first = true;
for(int x : xs) {
if (!first) write(' ');
first = false;
write(x);
}
return this;
}
public FastWriter write(long... xs)
{
boolean first = true;
for(long x : xs) {
if (!first) write(' ');
first = false;
write(x);
}
return this;
}
public FastWriter write(IntUnaryOperator f, int... xs)
{
boolean first = true;
for(int x : xs) {
if (!first) write(' ');
first = false;
write(f.applyAsInt(x));
}
return this;
}
public FastWriter write(LongUnaryOperator f, long... xs)
{
boolean first = true;
for(long x : xs) {
if (!first) write(' ');
first = false;
write(f.applyAsLong(x));
}
return this;
}
public FastWriter writeln()
{
return write((byte)'\n');
}
public FastWriter writeln(int... xs) { return write(xs).writeln(); }
public FastWriter writeln(long... xs) { return write(xs).writeln(); }
public FastWriter writeln(IntUnaryOperator f, int... xs) { return write(f, xs).writeln(); }
public FastWriter writeln(LongUnaryOperator f, long... xs) { return write(f, xs).writeln(); }
public FastWriter writeln(char[] line) { return write(line).writeln(); }
public FastWriter writeln(char[]... map) { for(char[] line : map)write(line).writeln();return this; }
public FastWriter writeln(String s) { return write(s).writeln(); }
private void innerflush()
{
try {
out.write(buf, 0, ptr);
ptr = 0;
} catch (IOException e) {
throw new RuntimeException("innerflush");
}
}
public void flush()
{
innerflush();
try {
out.flush();
} catch (IOException e) {
throw new RuntimeException("flush");
}
}
public FastWriter print(byte b) { return write(b); }
public FastWriter print(char c) { return write(c); }
public FastWriter print(char[] s) { return write(s); }
public FastWriter print(String s) { return write(s); }
public FastWriter print(int x) { return write(x); }
public FastWriter print(long x) { return write(x); }
public FastWriter print(double x, int precision) { return write(x, precision); }
public FastWriter println(char c){ return writeln(c); }
public FastWriter println(int x){ return writeln(x); }
public FastWriter println(long x){ return writeln(x); }
public FastWriter println(double x, int precision){ return writeln(x, precision); }
public FastWriter print(int... xs) { return write(xs); }
public FastWriter print(long... xs) { return write(xs); }
public FastWriter print(IntUnaryOperator f, int... xs) { return write(f, xs); }
public FastWriter print(LongUnaryOperator f, long... xs) { return write(f, xs); }
public FastWriter println(int... xs) { return writeln(xs); }
public FastWriter println(long... xs) { return writeln(xs); }
public FastWriter println(IntUnaryOperator f, int... xs) { return writeln(f, xs); }
public FastWriter println(LongUnaryOperator f, long... xs) { return writeln(f, xs); }
public FastWriter println(char[] line) { return writeln(line); }
public FastWriter println(char[]... map) { return writeln(map); }
public FastWriter println(String s) { return writeln(s); }
public FastWriter println() { return writeln(); }
}
public static void trnz(int... o)
{
for(int i = 0;i < o.length;i++)if(o[i] != 0)System.out.print(i+":"+o[i]+" ");
System.out.println();
}
// print ids which are 1
public static void trt(long... o)
{
Queue<Integer> stands = new ArrayDeque<>();
for(int i = 0;i < o.length;i++){
for(long x = o[i];x != 0;x &= x-1)stands.add(i<<6|Long.numberOfTrailingZeros(x));
}
System.out.println(stands);
}
public static void tf(boolean... r)
{
for(boolean x : r)System.out.print(x?'#':'.');
System.out.println();
}
public static void tf(boolean[]... b)
{
for(boolean[] r : b) {
for(boolean x : r)System.out.print(x?'#':'.');
System.out.println();
}
System.out.println();
}
public void tf(long[]... b)
{
if(INPUT.length() != 0) {
for (long[] r : b) {
for (long x : r) {
for (int i = 0; i < 64; i++) {
System.out.print(x << ~i < 0 ? '#' : '.');
}
}
System.out.println();
}
System.out.println();
}
}
public void tf(long... b)
{
if(INPUT.length() != 0) {
for (long x : b) {
for (int i = 0; i < 64; i++) {
System.out.print(x << ~i < 0 ? '#' : '.');
}
}
System.out.println();
}
}
private void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
}