#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pi;
typedef pair<ll, ll> pl;
typedef vector<int> vi;
typedef vector<ld> vd;
typedef vector<ll> vl;
#define pb push_back
#define f first
#define s second
namespace debug {
const int DEBUG = false;
template<class T1, class T2>
void pr(const pair<T1, T2> &x);
template<class T, size_t SZ>
void pr(const array<T, SZ> &x);
template<class T>
void pr(const vector<T> &x);
template<class T>
void pr(const set<T> &x);
template<class T1, class T2>
void pr(const map<T1, T2> &x);
template<class T>
void pr(const T &x) { if (DEBUG) cout << x; }
template<class T, class... Ts>
void pr(const T &first, const Ts &... rest) { pr(first), pr(rest...); }
template<class T1, class T2>
void pr(const pair<T1, T2> &x) { pr("{", x.f, ", ", x.s, "}"); }
template<class T>
void prIn(const T &x) {
pr("{");
bool fst = 1;
for (auto &a : x) {
pr(fst ? "" : ", ", a), fst = 0;
}
pr("}");
}
template<class T, size_t SZ>
void pr(const array<T, SZ> &x) { prIn(x); }
template<class T>
void pr(const vector<T> &x) { prIn(x); }
template<class T>
void pr(const set<T> &x) { prIn(x); }
template<class T1, class T2>
void pr(const map<T1, T2> &x) { prIn(x); }
void ps() { pr("\n"), cout << flush; }
template<class Arg, class... Args>
void ps(const Arg &first, const Args &... rest) {
pr(first, " ");
ps(rest...);
}
}
using namespace debug;
int N;
int K;
vi wSum;
bool canBlack(int l, int r) {
return (wSum[r] - (l == 0 ? 0 : wSum[l - 1])) == 0;
}
vector<vi> calcDP(string str, vi bS) {
wSum.resize(N);
wSum[0] = 1;
for (int i = 1; i < N; i++) {
wSum[i] = wSum[i - 1] + (str[i] == '_');
}
vector<vi> dp(N);
for (int i = 0; i < N; i++) {
dp[i].resize(K + 1, false);
}
dp[0][0] = true;
for (int i = 0; i + 1 < N; i++) {
for (int numF = 0; numF <= K; numF++) {
if (dp[i][numF] == false) {
continue;
}
// ps("consider:", i, numF);
if (str[i + 1] == 'X') {
if (numF == K) {
continue; // cannot lead to any valid combinations
}
int cBS = bS[numF];
// start a new block and end with white
if (i + 1 + cBS < N && canBlack(i + 1, i + cBS) && str[i + 1 + cBS] != 'X') {
dp[i + 1 + cBS][numF + 1] = true;
}
} else if (str[i + 1] == '_') {
dp[i + 1][numF] = true; // maintain status
} else {
// ps("3rd path");
assert(str[i + 1] == '.');
dp[i + 1][numF] = true; // maintain status
if (numF == K) {
continue;
}
int cBS = bS[numF]; // or start a new block
// ps(canBlack(i + 1, i + cBS), wSum[i + cBS] - wSum[i]);
if (i + 1 + cBS < N && canBlack(i + 1, i + cBS) && str[i + 1 + cBS] != 'X') {
dp[i + 1 + cBS][numF + 1] = true;
// ps(i, "->", i + 1 + cBS);
}
}
}
}
return dp;
}
string solve_puzzle(string str, vi bS) {
str = '_' + str + '_'; // white padding on sides
N = str.length();
K = bS.size();
string rStr(str.rbegin(), str.rend());
vi rBS = vi(bS.rbegin(), bS.rend());
// ps(str);
// ps(bS);
vector<vi> rInf = calcDP(rStr, rBS);
vector<vi> lInf = calcDP(str, bS);
reverse(rInf.begin(), rInf.end());
for (int i = 0; i < N; i++) {
reverse(rInf[i].begin(), rInf[i].end());
}
// ps(rInf);
//test if it can be black to see if must be white
vl dSum(N + 1, 0);
for (int i = 1; i < N; i++) { // block starts at i
for (int j = 0; j < K; j++) {
int rS = i + bS[j];
if (lInf[i - 1][j] && rS < N && rInf[rS][j + 1] && canBlack(i, rS - 1)) {
// ps(j, i, rS - 1);
dSum[i]++;
dSum[rS]--;
}
}
}
// ps("p1");
for (int i = 1; i < N; i++) {
dSum[i] += dSum[i - 1];
}
// ps(dSum, dSum);
string ans;
for (int i = 1; i < N - 1; i++) {
if (str[i] == '_') {
ans.pb('_');
} else if (str[i] == 'X') {
ans.pb('X');
} else if (dSum[i] == 0) {
ans.pb('_');
} else {
bool undet = false;
for (int j = 0; j <= K; j++) {
if (lInf[i][j] && rInf[i][j]) {
ans.pb('?');
undet = true;
break;
}
}
if (!undet) ans.pb('X');
}
}
return ans;
}
