/**
* 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