Problem:
Score : 1600 points

Problem Statement

Taichi thinks a binary string X of odd length N is beautiful if it is possible to apply the following operation \frac{N-1}{2} times so that the only character of the resulting string is 1 :

Choose three consecutive bits of X and replace them by their median. For example, we can turn 00110 into 010 by applying the operation to the middle three bits.

Taichi has a string S consisting of characters 0, 1 and ?. Taichi wants to know the number of ways to replace the question marks with 1 or 0 so that the resulting string is beautiful, modulo 10^{9} + 7.

Constraints

1 \leq |S| \leq 300000

|S| is odd.

All characters of S are either 0, 1 or ?.

Input

Input is given from Standard Input in the following format:

S

Output

Print the number of ways to replace the question marks so that the resulting string is beautiful, modulo 10^{9} + 7.

Sample Input 1

1??00

Sample Output 1

2

There are 4 ways to replace the question marks with 0 or 1 :

11100 : This string is beautiful because we can first perform the operation on the last 3 bits to get 110 and then on the only 3 bits to get 1.

11000 : This string is beautiful because we can first perform the operation on the last 3 bits to get 110 and then on the only 3 bits to get 1.

10100 : This string is not beautiful because there is no sequence of operations such that the final string is 1.

10000 : This string is not beautiful because there is no sequence of operations such that the final string is 1.

Thus, there are 2 ways to form a beautiful string.

Sample Input 2

?

Sample Output 2

1

In this case, 1 is the only beautiful string.

Sample Input 3

?0101???10???00?1???????????????0????????????1????0

Sample Output 3

402589311

Remember to output your answer modulo 10^{9} + 7.

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

// {{{ by shik
#include <bits/stdc++.h>
#pragma GCC diagnostic ignored "-Wunused-result"
#pragma GCC diagnostic ignored "-Wunused-function"
#define SZ(x) ((int)(x).size())
#define ALL(x) begin(x), end(x)
#define REP(i, n) for (int i = 0; i < int(n); i++)
#define REP1(i, a, b) for (int i = (a); i <= int(b); i++)
#define MP make_pair
#define PB push_back
using namespace std;
typedef int64_t LL;
typedef pair<int, int> PII;
typedef vector<int> VI;

namespace { namespace shik {

template<class T> void _R(T &x) { cin >> x; }
void _R(int &x) { scanf("%d", &x); }
void _R(int64_t &x) { scanf("%" SCNd64, &x); }
void _R(double &x) { scanf("%lf", &x); }
void _R(char &x) { scanf(" %c", &x); }
void _R(char *x) { scanf("%s", x); }
void R() {}
template<class T, class... U> void R(T &head, U &... tail) { _R(head); R(tail...); }

template<class T> void _W(const T &x) { cout << x; }
void _W(const int &x) { printf("%d", x); }
void _W(const int64_t &x) { printf("%" PRId64, x); }
void _W(const double &x) { printf("%.16f", x); }
void _W(const char &x) { putchar(x); }
void _W(const char *x) { printf("%s", x); }
template<class T> void _W(const vector<T> &x) { for (auto i = x.begin(); i != x.end(); _W(*i++)) if (i != x.cbegin()) putchar(' '); }
void W() {}
template<class T, class... U> void W(const T &head, const U &... tail) { _W(head); putchar(sizeof...(tail) ? ' ' : '\n'); W(tail...); }

#ifdef SHIK
#include "dump.hpp"
#else
#define dump(...)
#endif

template<class T, class F = less<T>> void sort_uniq(vector<T> &v, F f = F()) { sort(begin(v), end(v), f); v.resize(unique(begin(v), end(v)) - begin(v)); }
template<class T> inline T bit(T x, int i) { return (x >> i) & 1; }
template<class T> inline bool chkmax(T &a, const T &b) { return b > a ? a = b, true : false; }
template<class T> inline bool chkmin(T &a, const T &b) { return b < a ? a = b, true : false; }
template<class T> using MaxHeap = priority_queue<T>;
template<class T> using MinHeap = priority_queue<T, vector<T>, greater<T>>;

// }}}

// {{{ ModInt
template<int _MOD>
struct ModInt {
    static const auto MOD = _MOD;
    template<class T> using integral_only = typename enable_if<is_integral<T>::value>::type;

    int x;
    constexpr ModInt() : x() {}
    template<class T, integral_only<T>* = nullptr>
    ModInt(T _x) {
        x = _x % MOD;
        if (x < 0) x += MOD;
    }
    ModInt operator-() const { return {x == 0 ? 0 : MOD-x}; }
    ModInt& operator+=(ModInt rhs) {
        x += rhs.x;
        if (x >= MOD) x -= MOD;
        return *this;
    }
    ModInt& operator-=(ModInt rhs) {
        x -= rhs.x;
        if (x < 0) x += MOD;
        return *this;
    }
    ModInt& operator*=(ModInt rhs) {
        x = (long long)x * rhs.x % MOD;
        return *this;
    }
    ModInt& operator/=(ModInt rhs) {
        return *this *= rhs.inv();
    }
    ModInt operator+(ModInt rhs) const { return ModInt(*this) += rhs; }
    ModInt operator-(ModInt rhs) const { return ModInt(*this) -= rhs; }
    ModInt operator*(ModInt rhs) const { return ModInt(*this) *= rhs; }
    ModInt operator/(ModInt rhs) const { return ModInt(*this) /= rhs; }
    ModInt inv() const {
        // should work for non-prime MOD if gcd(x, MOD) = 1
        int a = x, b = MOD, u = 1, v = 0;
        while (b != 0) {
            int t = a / b;
            a -= t * b;
            u -= t * v;
            swap(a, b);
            swap(u, v);
        }
        return u;
    }
    template<class T, integral_only<T>* = nullptr>
    ModInt pow(T e) {
        ModInt r = 1, p = *this;
        while (e) {
            if (e & 1) r *= p;
            p *= p;
            e >>= 1;
        }
        return r;
    }
    bool operator==(ModInt rhs) const { return x == rhs.x; }
    bool operator!=(ModInt rhs) const { return x != rhs.x; }
    bool operator<(ModInt rhs) const { return x < rhs.x; }
    bool operator<=(ModInt rhs) const { return x <= rhs.x; }
    bool operator>(ModInt rhs) const { return x > rhs.x; }
    bool operator>=(ModInt rhs) const { return x >= rhs.x; }
    friend string to_string(ModInt i) { return to_string(i.x); }
    friend ostream& operator<<(ostream &os, ModInt i) { return os << i.x; }
};
// }}}

const int MOD = 1e9 + 7;
using mint = ModInt<MOD>;

const int N=3e5+10;
const int M=10;
int n;
char s[N];
mint _dp[2][2*M+1][3];
auto dp=_dp[0]+M,dq=_dp[1]+M;

enum {
    Z,
    Z0,
    Z00,
};

const int to[3][2][2]={
    {{Z0,-1},{Z,+1}},
    {{Z00,-1},{Z,+1}},
    {{Z0,+1},{Z0,+1}},
};

string bin( int x, int l ) {
    string t;
    REP(i,l) {
        t.push_back('0' + x%2);
        x/=2;
    }
    return t;
}

int slow() {
    int ans=0;
    REP(i,1<<n) {
        bool ok=1;
        REP(j,n) ok&=s[j]=='?' || (s[j]-'0' == ((i>>j)&1));
        if ( !ok ) continue;
        int c=0,z=Z;
        REP(j,n) {
            auto t=to[z][(i>>j)&1];
            z=t[0];
            c+=t[1];
        }
        if ( c>0 ) {
            ans++;
            // dump(bin(i,n), i);
        } else {
            dump(bin(i,n), i);
        }
    }
    return ans;
}

mint p2[N];
int q[N];
void main() {
    R(s);
    n=strlen(s);

    p2[0]=1;
    REP1(i,1,n) p2[i]=p2[i-1]+p2[i-1];

    for ( int i=n-1; i>=0; i-- ) q[i]=q[i+1]+(s[i]=='?');

    mint ans=0;
    dp[0][Z]=1;
    REP(i,n) {
        // REP1(j,-2,+5) REP(k,3) if ( dp[j][k]!=0 ) dump(i,j,k,dp[j][k]);
        REP1(j,-2,+6) REP(k,3) dq[j][k]=0;
        if ( s[i]=='0' || s[i]=='?' ) {
            REP1(j,-2,+5) REP(k,3) {
                auto t=to[k][0];
                dq[j+t[1]][t[0]]+=dp[j][k];
            }
        }
        if ( s[i]=='1' || s[i]=='?' ) {
            REP1(j,-2,+5) REP(k,3) {
                auto t=to[k][1];
                dq[j+t[1]][t[0]]+=dp[j][k];
            }
        }
        REP(j,3) ans+=dq[6][j]*p2[q[i+1]];
        swap(dp,dq);
    }
    REP1(i,1,+5) REP(j,3) ans+=dp[i][j];
    W(ans);
#ifdef SHIK
    if ( n<20 ) {
        int ans2=slow();
        dump(ans,ans2);
        assert(ans==ans2);
    }
#endif
}

// {{{ main
}}
int main() { shik::main(); return 0; }
// }}}