#include <cstdio>
#include <iostream>
#include <cstring>
#include <algorithm>
#include <vector>
#include <map>
#include <string>
#define PB push_back
#define MP make_pair
#define PII pair<int,int>
#define FIR first
#define SEC second
#define ll long long
using namespace std;
template <class T>
inline void rd(T &x)
{
    x = 0;
    char c = getchar();
    int f = 1;

    while (!isdigit(c))
    {
        if (c == '-')
            f = -1;

        c = getchar();
    }

    while (isdigit(c))
        x = x * 10 - '0' + c, c = getchar();

    x *= f;
}
inline void rdS(string &S)
{
    S = "";
    char c = getchar();

    while (c != '0' && c != '1')
        c = getchar();

    while (c == '0' || c == '1')
        S += c, c = getchar();
}
const int mod = 1e9 + 9;
map<ll, int> dp;
ll calcdp(ll k)
{
    if (dp.count(k))
        return dp[k];

    if (k == 0)
        return dp[k] = 1;

    if (k == 1)
        return dp[k] = 2;

    if (k == 2)
        return dp[k] = 3;

    calcdp(k + 1 >> 1), calcdp(k >> 1);
    return dp[k] = (dp[k + 1 >> 1] + dp[k >> 1]) % mod;
}
ll calcans(string S, ll k)
{
    if (S.length() == 1)
        return calcdp(k);

    if (k + S.length() <= 3)
    {
        if (k == 0)
        {
            if (S == "000" || S == "111")
                return 0;

            return 1;
        }

        if (k == 1)
        {
            if (S == "00" || S == "11")
                return 1;

            return 2;
        }

        return calcdp(k);
    }

    string T = "";
    ll res = 0;
    int flg = 1;

    for (int i = 0; i < S.length() && flg; i += 2)
    {
        if (i + 1 < S.length() && S[i] == S[i + 1])
            flg = 0;

        T += S[i];
    }

    if (flg /*&& k-(S.length()&1)>=0*/)
        res = (res + calcans(T, k - (S.length() & 1) + 1 >> 1)) % mod;

    flg = 1;
    T = "";
    T += ('0' + '1' - S[0]);

    for (int i = 1; i < S.length() && flg; i += 2)
    {
        if (i + 1 < S.length() && S[i] == S[i + 1])
            flg = 0;

        T += S[i];
    }

    if (flg /*&& k-(S.length()-1&1)>=0*/)
        res = (res + calcans(T, k - (S.length() - 1 & 1) + 1 >> 1)) % mod;

    return res;
}
string S;
ll k;
int main()
{
    ios::sync_with_stdio(false);
    int T;// rd(T);
    cin >> T;

    while (T--)
    {
        //      rdS(S),rd(k);
        //      printf("%lld\n",calcans(S,k));
        cin >> S >> k;
        cout << calcans(S, k) << endl;
        dp.clear();
    }

    return 0;
}