結果
問題 | No.1962 Not Divide |
ユーザー |
|
提出日時 | 2022-06-09 18:21:00 |
言語 | Java (openjdk 23) |
結果 |
TLE
|
実行時間 | - |
コード長 | 32,408 bytes |
コンパイル時間 | 5,867 ms |
コンパイル使用メモリ | 94,804 KB |
実行使用メモリ | 107,036 KB |
最終ジャッジ日時 | 2024-09-21 05:37:11 |
合計ジャッジ時間 | 44,760 ms |
ジャッジサーバーID (参考情報) |
judge2 / judge4 |
(要ログイン)
ファイルパターン | 結果 |
---|---|
sample | AC * 3 |
other | AC * 11 TLE * 10 |
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++){long[] f = new long[D];for(int j = 0;j < D;j++){if(j % i != 0){f[j] = 1;}}f[0]++;r = sub(r, inv(f));}// tr(r);// H/(H+1) = r// H = (H+1)r// H = 1/(1-r)-1long[] 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 lengthlong[] 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);// }// normalizelong 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 thresholdif(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 998244353public 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;}// differentiatepublic 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;}// integratepublic 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 1public 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)); }}