Problem:
Score : 1700 points

Problem Statement

You have two strings A = A_1 A_2 ... A_n and B = B_1 B_2 ... B_n of the same length consisting of 0 and 1.
The number of 1's in A and B is equal.

You've decided to transform A using the following algorithm:

Let a_1, a_2, ..., a_k be the indices of 1's in A.

Let b_1, b_2, ..., b_k be the indices of 1's in B.

Replace a and b with their random permutations, chosen independently and uniformly.

For each i from 1 to k, in order, swap A_{a_i} and A_{b_i}.

Let P be the probability that strings A and B become equal after the procedure above.

Let Z = P \times (k!)^2. Clearly, Z is an integer.

Find Z modulo 998244353.

Constraints

1 \leq |A| = |B| \leq 10,000

A and B consist of 0 and 1.

A and B contain the same number of 1's.

A and B contain at least one 1.

Partial Score

1200 points will be awarded for passing the testset satisfying 1 \leq |A| = |B| \leq 500.

Input

Input is given from Standard Input in the following format:

A
B

Output

Print the value of Z modulo 998244353.

Sample Input 1

1010
1100

Sample Output 1

3

After the first two steps, a = [1, 3] and b = [1, 2]. There are 4 possible scenarios after shuffling a and b:

a = [1, 3], b = [1, 2]. Initially, A = 1010. After swap(A_1, A_1), A = 1010. After swap(A_3, A_2), A = 1100.

a = [1, 3], b = [2, 1]. Initially, A = 1010. After swap(A_1, A_2), A = 0110. After swap(A_3, A_1), A = 1100.

a = [3, 1], b = [1, 2]. Initially, A = 1010. After swap(A_3, A_1), A = 1010. After swap(A_1, A_2), A = 0110.

a = [3, 1], b = [2, 1]. Initially, A = 1010. After swap(A_3, A_2), A = 1100. After swap(A_1, A_1), A = 1100.

Out of 4 scenarios, 3 of them result in A = B. Therefore, P = 3 / 4, and Z = 3.

Sample Input 2

01001
01001

Sample Output 2

4

No swap ever changes A, so we'll always have A = B.

Sample Input 3

101010
010101

Sample Output 3

36

Every possible sequence of three swaps puts the 1's in A into the right places.

Sample Input 4

1101011011110
0111101011101

Sample Output 4

932171449

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;

public class Main {
	static InputStream is;
	static PrintWriter out;
	static String INPUT = "";
	
	static void solve()
	{
		char[] a = ns().toCharArray();
		char[] b = ns().toCharArray();
		int o = 0, d = 0;
		for(int i = 0;i < a.length;i++){
			if(a[i] == b[i] && a[i] == '1'){
				d++;
				continue;
			}
			if(a[i] == b[i] && a[i] == '0')continue;
			o++;
		}
		o /= 2;
		int k = o + d;
		int mod = 998244353;
		long[] dp = new long[d+1];
		int[][] fif = enumFIF(100000, mod);
		long[] u = new long[d+1];
		for(int i = 0;i <= d;i++){
			u[i] = fif[1][i+1];
		}
		dp[0] = 1;
		
		for(int z = 1;z <= o;z*=2){
			if((o&z) > 0){
				dp = Arrays.copyOf(convoluteSimply(dp, u, mod, 3), d+1);
			}
			u = Arrays.copyOf(convoluteSimply(u, u, mod, 3), d+1);
		}
		
		// 
		long ans = 0;
		for(int i = 0;i <= d;i++){
//			ans += fif[0][d] * dp[i] % mod * fif[1][d-i] % mod * fif[0][d-i] % mod * fif[0][o] % mod * fif[0][k] % mod;
			ans += dp[i];
			ans %= mod;
		}
		out.println(ans * fif[0][d] % mod * fif[0][o] % mod * fif[0][k] % mod);
	}
	
	
	public static final int[] NTTPrimes = {1053818881, 1051721729, 1045430273, 1012924417, 1007681537, 1004535809, 998244353, 985661441, 976224257, 975175681};
	public static final int[] NTTPrimitiveRoots = {7, 6, 3, 5, 3, 3, 3, 3, 3, 17};
//	public static final int[] NTTPrimes = {1012924417, 1004535809, 998244353, 985661441, 975175681, 962592769, 950009857, 943718401, 935329793, 924844033};
//	public static final int[] NTTPrimitiveRoots = {5, 3, 3, 3, 17, 7, 7, 7, 3, 5};
	
	public static long[] convoluteSimply(long[] a, long[] b, int P, int g)
	{
		int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
		long[] fa = nttmb(a, m, false, P, g);
		long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
		for(int i = 0;i < m;i++){
			fa[i] = fa[i]*fb[i]%P;
		}
		return nttmb(fa, m, true, P, g);
	}
	
	public static long[] convolute(long[] a, long[] b)
	{
		int USE = 2;
		int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
		long[][] fs = new long[USE][];
		for(int k = 0;k < USE;k++){
			int P = NTTPrimes[k], g = NTTPrimitiveRoots[k];
			long[] fa = nttmb(a, m, false, P, g);
			long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
			for(int i = 0;i < m;i++){
				fa[i] = fa[i]*fb[i]%P;
			}
			fs[k] = nttmb(fa, m, true, P, g);
		}
		
		int[] mods = Arrays.copyOf(NTTPrimes, USE);
		long[] gammas = garnerPrepare(mods);
		int[] buf = new int[USE];
		for(int i = 0;i < fs[0].length;i++){
			for(int j = 0;j < USE;j++)buf[j] = (int)fs[j][i];
			long[] res = garnerBatch(buf, mods, gammas);
			long ret = 0;
			for(int j = res.length-1;j >= 0;j--)ret = ret * mods[j] + res[j];
			fs[0][i] = ret;
		}
		return fs[0];
	}
	
	public static long[] convolute(long[] a, long[] b, int USE, int mod)
	{
		int m = Math.max(2, Integer.highestOneBit(Math.max(a.length, b.length)-1)<<2);
		long[][] fs = new long[USE][];
		for(int k = 0;k < USE;k++){
			int P = NTTPrimes[k], g = NTTPrimitiveRoots[k];
			long[] fa = nttmb(a, m, false, P, g);
			long[] fb = a == b ? fa : nttmb(b, m, false, P, g);
			for(int i = 0;i < m;i++){
				fa[i] = fa[i]*fb[i]%P;
			}
			fs[k] = nttmb(fa, m, true, P, g);
		}
		
		int[] mods = Arrays.copyOf(NTTPrimes, USE);
		long[] gammas = garnerPrepare(mods);
		int[] buf = new int[USE];
		for(int i = 0;i < fs[0].length;i++){
			for(int j = 0;j < USE;j++)buf[j] = (int)fs[j][i];
			long[] res = garnerBatch(buf, mods, gammas);
			long ret = 0;
			for(int j = res.length-1;j >= 0;j--)ret = (ret * mods[j] + res[j]) % mod;
			fs[0][i] = ret;
		}
		return fs[0];
	}
	
	// static int[] wws = new int[270000]; // outer faster
	
	// Modifed Montgomery + Barrett
	private static long[] nttmb(long[] src, int n, boolean inverse, int P, int g)
	{
		long[] dst = Arrays.copyOf(src, n);
		
		int h = Integer.numberOfTrailingZeros(n);
		long K = Integer.highestOneBit(P)<<1;
		int H = Long.numberOfTrailingZeros(K)*2;
		long M = K*K/P;
		
		int[] wws = new int[1<<h-1];
		long dw = inverse ? pow(g, P-1-(P-1)/n, P) : pow(g, (P-1)/n, P);
		long w = (1L<<32)%P;
		for(int k = 0;k < 1<<h-1;k++){
			wws[k] = (int)w;
			w = modh(w*dw, M, H, P);
		}
		long J = invl(P, 1L<<32);
		for(int i = 0;i < h;i++){
			for(int j = 0;j < 1<<i;j++){
				for(int k = 0, s = j<<h-i, t = s|1<<h-i-1;k < 1<<h-i-1;k++,s++,t++){
					long u = (dst[s] - dst[t] + 2*P)*wws[k];
					dst[s] += dst[t];
					if(dst[s] >= 2*P)dst[s] -= 2*P;
//					long Q = (u&(1L<<32)-1)*J&(1L<<32)-1;
					long Q = (u<<32)*J>>>32;
					dst[t] = (u>>>32)-(Q*P>>>32)+P;
				}
			}
			if(i < h-1){
				for(int k = 0;k < 1<<h-i-2;k++)wws[k] = wws[k*2];
			}
		}
		for(int i = 0;i < n;i++){
			if(dst[i] >= P)dst[i] -= P;
		}
		for(int i = 0;i < n;i++){
			int rev = Integer.reverse(i)>>>-h;
			if(i < rev){
				long d = dst[i]; dst[i] = dst[rev]; dst[rev] = d;
			}
		}
		
		if(inverse){
			long in = invl(n, P);
			for(int i = 0;i < n;i++)dst[i] = modh(dst[i]*in, M, H, P);
		}
		
		return dst;
	}
	
	// Modified Shoup + Barrett
	private static long[] nttsb(long[] src, int n, boolean inverse, int P, int g)
	{
		long[] dst = Arrays.copyOf(src, n);
		
		int h = Integer.numberOfTrailingZeros(n);
		long K = Integer.highestOneBit(P)<<1;
		int H = Long.numberOfTrailingZeros(K)*2;
		long M = K*K/P;
		
		long dw = inverse ? pow(g, P-1-(P-1)/n, P) : pow(g, (P-1)/n, P);
		long[] wws = new long[1<<h-1];
		long[] ws = new long[1<<h-1];
		long w = 1;
		for(int k = 0;k < 1<<h-1;k++){
			wws[k] = (w<<32)/P;
			ws[k] = w;
			w = modh(w*dw, M, H, P);
		}
		for(int i = 0;i < h;i++){
			for(int j = 0;j < 1<<i;j++){
				for(int k = 0, s = j<<h-i, t = s|1<<h-i-1;k < 1<<h-i-1;k++,s++,t++){
					long ndsts = dst[s] + dst[t];
					if(ndsts >= 2*P)ndsts -= 2*P;
					long T = dst[s] - dst[t] + 2*P;
					long Q = wws[k]*T>>>32;
					dst[s] = ndsts;
					dst[t] = ws[k]*T-Q*P&(1L<<32)-1;
				}
			}
//			dw = dw * dw % P;
			if(i < h-1){
				for(int k = 0;k < 1<<h-i-2;k++){
					wws[k] = wws[k*2];
					ws[k] = ws[k*2];
				}
			}
		}
		for(int i = 0;i < n;i++){
			if(dst[i] >= P)dst[i] -= P;
		}
		for(int i = 0;i < n;i++){
			int rev = Integer.reverse(i)>>>-h;
			if(i < rev){
				long d = dst[i]; dst[i] = dst[rev]; dst[rev] = d;
			}
		}
		
		if(inverse){
			long in = invl(n, P);
			for(int i = 0;i < n;i++){
				dst[i] = modh(dst[i] * in, M, H, P);
			}
		}
		
		return dst;
	}
	
	static final long mask = (1L<<31)-1;
	
	public static long modh(long a, long M, int h, int mod)
	{
		long r = a-((M*(a&mask)>>>31)+M*(a>>>31)>>>h-31)*mod;
		return r < mod ? r : r-mod;
	}
	
	private static long[] garnerPrepare(int[] m)
	{
		int n = m.length;
		assert n == m.length;
		if(n == 0)return new long[0];
		long[] gamma = new long[n];
		for(int k = 1;k < n;k++){
			long prod = 1;
			for(int i = 0;i < k;i++){
				prod = prod * m[i] % m[k];
			}
			gamma[k] = invl(prod, m[k]);
		}
		return gamma;
	}
	
	private static long[] garnerBatch(int[] u, int[] m, long[] gamma)
	{
		int n = u.length;
		assert n == m.length;
		long[] v = new long[n];
		v[0] = u[0];
		for(int k = 1;k < n;k++){
			long temp = v[k-1];
			for(int j = k-2;j >= 0;j--){
				temp = (temp * m[j] + v[j]) % m[k];
			}
			v[k] = (u[k] - temp) * gamma[k] % m[k];
			if(v[k] < 0)v[k] += m[k];
		}
		return v;
	}
	
	private 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;
	}
	
	private 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 long C(int n, int r, int mod, int[][] fif) {
		if (n < 0 || r < 0 || r > n)
			return 0;
		return (long) fif[0][n] * fif[1][r] % mod * fif[1][n - r] % mod;
	}

	
	public static int[][] enumFIF(int n, int mod) {
		int[] f = new int[n + 1];
		int[] invf = new int[n + 1];
		f[0] = 1;
		for (int i = 1; i <= n; i++) {
			f[i] = (int) ((long) f[i - 1] * i % mod);
		}
		long a = f[n];
		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;
		}
		invf[n] = (int) (p < 0 ? p + mod : p);
		for (int i = n - 1; i >= 0; i--) {
			invf[i] = (int) ((long) invf[i + 1] * (i + 1) % mod);
		}
		return new int[][] { f, invf };
	}

	
	public static void main(String[] args) throws Exception
	{
		long S = System.currentTimeMillis();
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		solve();
		out.flush();
		long G = System.currentTimeMillis();
		tr(G-S+"ms");
	}
	
	private static 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 static byte[] inbuf = new byte[1024];
	static int lenbuf = 0, ptrbuf = 0;
	
	private static 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 static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
//	private static boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private static double nd() { return Double.parseDouble(ns()); }
	private static char nc() { return (char)skip(); }
	
	private static String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private static 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 static 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 static int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private static 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 static 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();
		}
	}
	
	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
}