/**

* Convolution.

*

* @verified https://atcoder.jp/contests/practice2/submissions/24449847

* @verified https://judge.yosupo.jp/submission/53841

*/

class Convolution {

/**

* Find a primitive root.

*

* @param m A prime number.

* @return Primitive root.

*/

private static int primitiveRoot(int m) {

if (m == 2) return 1;

if (m == 167772161) return 3;

if (m == 469762049) return 3;

if (m == 754974721) return 11;

if (m == 998244353) return 3;

int[] divs = new int[20];

divs[0] = 2;

int cnt = 1;

int x = (m - 1) / 2;

while (x % 2 == 0) x /= 2;

for (int i = 3; (long) (i) * i <= x; i += 2) {

if (x % i == 0) {

divs[cnt++] = i;

while (x % i == 0) {

x /= i;

}

}

}

if (x > 1) {

divs[cnt++] = x;

}

for (int g = 2; ; g++) {

boolean ok = true;

for (int i = 0; i < cnt; i++) {

if (pow(g, (m - 1) / divs[i], m) == 1) {

ok = false;

break;

}

}

if (ok) return g;

}

}

/**

* Power.

*

* @param x Parameter x.

* @param n Parameter n.

* @param m Mod.

* @return n-th power of x mod m.

*/

private static long pow(long x, long n, int m) {

if (m == 1) return 0;

long r = 1;

long y = x % m;

while (n > 0) {

if ((n & 1) != 0) r = (r * y) % m;

y = (y * y) % m;

n >>= 1;

}

return r;

}

/**

* Ceil of power 2.

*

* @param n Value.

* @return Ceil of power 2.

*/

private static int ceilPow2(int n) {

int x = 0;

while ((1L << x) < n) x++;

return x;

}

private static class FftInfo {

private static int bsfConstexpr(int n) {

int x = 0;

while ((n & (1 << x)) == 0) x++;

return x;

}

private static long inv(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;

}

private final int rank2;

public final long[] root;

public final long[] iroot;

public final long[] rate2;

public final long[] irate2;

public final long[] rate3;

public final long[] irate3;

public FftInfo(int g, int mod) {

rank2 = bsfConstexpr(mod - 1);

root = new long[rank2 + 1];

iroot = new long[rank2 + 1];

rate2 = new long[Math.max(0, rank2 - 2 + 1)];

irate2 = new long[Math.max(0, rank2 - 2 + 1)];

rate3 = new long[Math.max(0, rank2 - 3 + 1)];

irate3 = new long[Math.max(0, rank2 - 3 + 1)];

root[rank2] = pow(g, (mod - 1) >> rank2, mod);

iroot[rank2] = inv(root[rank2], mod);

for (int i = rank2 - 1; i >= 0; i--) {

root[i] = root[i + 1] * root[i + 1] % mod;

iroot[i] = iroot[i + 1] * iroot[i + 1] % mod;

}

{

long prod = 1, iprod = 1;

for (int i = 0; i <= rank2 - 2; i++) {

rate2[i] = root[i + 2] * prod % mod;

irate2[i] = iroot[i + 2] * iprod % mod;

prod = prod * iroot[i + 2] % mod;

iprod = iprod * root[i + 2] % mod;

}

}

{

long prod = 1, iprod = 1;

for (int i = 0; i <= rank2 - 3; i++) {

rate3[i] = root[i + 3] * prod % mod;

irate3[i] = iroot[i + 3] * iprod % mod;

prod = prod * iroot[i + 3] % mod;

iprod = iprod * root[i + 3] % mod;

}

}

}

};

/**

* Garner's algorithm.

*

* @param c Mod convolution results.

* @param mods Mods.

* @return Result.

*/

private static long garner(long[] c, int[] mods) {

int n = c.length + 1;

long[] cnst = new long[n];

long[] coef = new long[n];

java.util.Arrays.fill(coef, 1);

for (int i = 0; i < n - 1; i++) {

int m1 = mods[i];

long v = (c[i] - cnst[i] + m1) % m1;

v = v * pow(coef[i], m1 - 2, m1) % m1;

for (int j = i + 1; j < n; j++) {

long m2 = mods[j];

cnst[j] = (cnst[j] + coef[j] * v) % m2;

coef[j] = (coef[j] * m1) % m2;

}

}

return cnst[n - 1];

}

/**

* Inverse NTT.

*

* @param a Target array.

* @param g Primitive root of mod.

* @param mod NTT Prime.

*/

private static void butterflyInv(long[] a, int g, int mod) {

int n = a.length;

int h = ceilPow2(n);

FftInfo info = new FftInfo(g, mod);

int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed

while (len > 0) {

if (len == 1) {

int p = 1 << (h - len);

long irot = 1;

for (int s = 0; s < (1 << (len - 1)); s++) {

int offset = s << (h - len + 1);

for (int i = 0; i < p; i++) {

long l = a[i + offset];

long r = a[i + offset + p];

a[i + offset] = (l + r) % mod;

a[i + offset + p] = (mod + l - r) % mod * irot % mod;

}

if (s + 1 != (1 << (len - 1))) {

irot *= info.irate2[Integer.numberOfTrailingZeros(~s)];

irot %= mod;

}

}

len--;

} else {

// 4-base

int p = 1 << (h - len);

long irot = 1, iimag = info.iroot[2];

for (int s = 0; s < (1 << (len - 2)); s++) {

long irot2 = irot * irot % mod;

long irot3 = irot2 * irot % mod;

int offset = s << (h - len + 2);

for (int i = 0; i < p; i++) {

long a0 = 1L * a[i + offset + 0 * p];

long a1 = 1L * a[i + offset + 1 * p];

long a2 = 1L * a[i + offset + 2 * p];

long a3 = 1L * a[i + offset + 3 * p];

long a2na3iimag = 1L * (mod + a2 - a3) % mod * iimag % mod;

a[i + offset] = (a0 + a1 + a2 + a3) % mod;

a[i + offset + 1 * p] = (a0 + (mod - a1) + a2na3iimag) % mod * irot % mod;

a[i + offset + 2 * p] = (a0 + a1 + (mod - a2) + (mod - a3)) % mod * irot2 % mod;

a[i + offset + 3 * p] = (a0 + (mod - a1) + (mod - a2na3iimag)) % mod * irot3 % mod;

}

if (s + 1 != (1 << (len - 2))) {

irot *= info.irate3[Integer.numberOfTrailingZeros(~s)];

irot %= mod;

}

}

len -= 2;

}

}

}

/**

* Inverse NTT.

*

* @param a Target array.

* @param g Primitive root of mod.

* @param mod NTT Prime.

*/

private static void butterfly(long[] a, int g, int mod) {

int n = a.length;

int h = ceilPow2(n);

FftInfo info = new FftInfo(g, mod);

int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed

while (len < h) {

if (h - len == 1) {

int p = 1 << (h - len - 1);

long rot = 1;

for (int s = 0; s < (1 << len); s++) {

int offset = s << (h - len);

for (int i = 0; i < p; i++) {

long l = a[i + offset];

long r = a[i + offset + p] * rot % mod;

a[i + offset] = (l + r) % mod;

a[i + offset + p] = (l + mod - r) % mod;

}

if (s + 1 != (1 << len)) {

rot *= info.rate2[Integer.numberOfTrailingZeros(~s)];

rot %= mod;

}

}

len++;

} else {

// 4-base

int p = 1 << (h - len - 2);

long rot = 1, imag = info.root[2];

for (int s = 0; s < (1 << len); s++) {

long rot2 = rot * rot % mod;

long rot3 = rot2 * rot % mod;

int offset = s << (h - len);

for (int i = 0; i < p; i++) {

long mod2 = 1L * mod * mod;

long a0 = 1L * a[i + offset];

long a1 = 1L * a[i + offset + p] * rot % mod;

long a2 = 1L * a[i + offset + 2 * p] * rot2 % mod;

long a3 = 1L * a[i + offset + 3 * p] * rot3 % mod;

long a1na3imag = 1L * (a1 + mod2 - a3) % mod * imag % mod;

long na2 = mod2 - a2;

a[i + offset] = (a0 + a2 + a1 + a3) % mod;

a[i + offset + 1 * p] = (a0 + a2 + (2 * mod2 - (a1 + a3))) % mod;

a[i + offset + 2 * p] = (a0 + na2 + a1na3imag) % mod;

a[i + offset + 3 * p] = (a0 + na2 + (mod2 - a1na3imag)) % mod;

}

if (s + 1 != (1 << len)) {

rot *= info.rate3[Integer.numberOfTrailingZeros(~s)];

rot %= mod;

}

}

len += 2;

}

}

}

/**

* Convolution.

*

* @param a Target array 1.

* @param b Target array 2.

* @param mod NTT Prime.

* @return Answer.

*/

public static long[] convolution(long[] a, long[] b, int mod) {

int n = a.length;

int m = b.length;

if (n == 0 || m == 0) return new long[0];

int z = 1 << ceilPow2(n + m - 1);

{

long[] na = new long[z];

long[] nb = new long[z];

System.arraycopy(a, 0, na, 0, n);

System.arraycopy(b, 0, nb, 0, m);

a = na;

b = nb;

}

int g = primitiveRoot(mod);

butterfly(a, g, mod);

butterfly(b, g, mod);

for (int i = 0; i < z; i++) {

a[i] = a[i] * b[i] % mod;

}

butterflyInv(a, g, mod);

a = java.util.Arrays.copyOf(a, n + m - 1);

long iz = pow(z, mod - 2, mod);

for (int i = 0; i < n + m - 1; i++) a[i] = a[i] * iz % mod;

return a;

}

/**

* Convolution.

*

* @param a Target array 1.

* @param b Target array 2.

* @param mod Any mod.

* @return Answer.

*/

public static long[] convolutionLL(long[] a, long[] b, int mod) {

int n = a.length;

int m = b.length;

if (n == 0 || m == 0) return new long[0];

int mod1 = 754974721;

int mod2 = 167772161;

int mod3 = 469762049;

long[] c1 = convolution(a, b, mod1);

long[] c2 = convolution(a, b, mod2);

long[] c3 = convolution(a, b, mod3);

int retSize = c1.length;

long[] ret = new long[retSize];

int[] mods = {mod1, mod2, mod3, mod};

for (int i = 0; i < retSize; ++i) {

ret[i] = garner(new long[]{c1[i], c2[i], c3[i]}, mods);

}

return ret;

}

/**

* Convolution by ModInt.

*

* @param a Target array 1.

* @param b Target array 2.

* @return Answer.

*/

public static java.util.List<ModIntFactory.ModInt> convolution(

java.util.List<ModIntFactory.ModInt> a,

java.util.List<ModIntFactory.ModInt> b

) {

int mod = a.get(0).mod();

long[] va = a.stream().mapToLong(ModIntFactory.ModInt::value).toArray();

long[] vb = b.stream().mapToLong(ModIntFactory.ModInt::value).toArray();

long[] c = convolutionLL(va, vb, mod);

ModIntFactory factory = new ModIntFactory(mod);

return java.util.Arrays.stream(c).mapToObj(factory::create).collect(java.util.stream.Collectors.toList());

}

/**

* Naive convolution. (Complexity is O(N^2)!!)

*

* @param a Target array 1.

* @param b Target array 2.

* @param mod Mod.

* @return Answer.

*/

public static long[] convolutionNaive(long[] a, long[] b, int mod) {

int n = a.length;

int m = b.length;

int k = n + m - 1;

long[] ret = new long[k];

for (int i = 0; i < n; i++) {

for (int j = 0; j < m; j++) {

ret[i + j] += a[i] * b[j] % mod;

ret[i + j] %= mod;

}

}

return ret;

}

}