#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include<iostream>
#include<string>
#include<cstdio>
#include<vector>
#include<cmath>
#include<algorithm>
#include<functional>
#include<iomanip>
#include<queue>
#include<ciso646>
#include<random>
#include<map>
#include<set>
#include<bitset>
#include<stack>
#include<unordered_map>
#include<unordered_set>
#include<utility>
#include<cassert>
#include<complex>
#include<numeric>
#include<array>
#include<chrono>
using namespace std;

//#define int long long
typedef long long ll;

typedef unsigned long long ul;
typedef unsigned int ui;
//ll mod = 1;
//constexpr ll mod = 998244353;
constexpr ll mod = 1000000007;
const int mod17 = 1000000007;
const ll INF = (ll)mod17 * mod17;
typedef pair<int, int>P;

#define rep(i,n) for(int i=0;i<n;i++)
#define per(i,n) for(int i=n-1;i>=0;i--)
#define Rep(i,sta,n) for(int i=sta;i<n;i++)
#define rep1(i,n) for(int i=1;i<=n;i++)
#define per1(i,n) for(int i=n;i>=1;i--)
#define Rep1(i,sta,n) for(int i=sta;i<=n;i++)
#define all(v) (v).begin(),(v).end()
typedef pair<ll, ll> LP;

using ld = double;
typedef pair<ld, ld> LDP;
const ld eps = 1e-10;
const ld pi = acosl(-1.0);

template<typename T>
void chmin(T& a, T b) {
    a = min(a, b);
}
template<typename T>
void chmax(T& a, T b) {
    a = max(a, b);
}
template<typename T>
vector<T> vmerge(vector<T>& a, vector<T>& b) {
    vector<T> res;
    int ida = 0, idb = 0;
    while (ida < a.size() || idb < b.size()) {
        if (idb == b.size()) {
            res.push_back(a[ida]); ida++;
        }
        else if (ida == a.size()) {
            res.push_back(b[idb]); idb++;
        }
        else {
            if (a[ida] < b[idb]) {
                res.push_back(a[ida]); ida++;
            }
            else {
                res.push_back(b[idb]); idb++;
            }
        }
    }
    return res;
}
template<typename T>
void cinarray(vector<T>& v) {
    rep(i, v.size())cin >> v[i];
}
template<typename T>
void coutarray(vector<T>& v) {
    rep(i, v.size()) {
        if (i > 0)cout << " "; cout << v[i];
    }
    cout << "\n";
}
ll mod_pow(ll x, ll n, ll m = mod) {
    if (n < 0) {
        ll res = mod_pow(x, -n, m);
        return mod_pow(res, m - 2, m);
    }
    if (abs(x) >= m)x %= m;
    if (x < 0)x += m;
    //if (x == 0)return 0;
    ll res = 1;
    while (n) {
        if (n & 1)res = res * x % m;
        x = x * x % m; n >>= 1;
    }
    return res;
}
//mod should be <2^31
struct modint {
    int n;
    modint() :n(0) { ; }
    modint(ll m) {
        if (m < 0 || mod <= m) {
            m %= mod; if (m < 0)m += mod;
        }
        n = m;
    }
    operator int() { return n; }
};
bool operator==(modint a, modint b) { return a.n == b.n; }
bool operator<(modint a, modint b) { return a.n < b.n; }
modint operator+=(modint& a, modint b) { a.n += b.n; if (a.n >= mod)a.n -= (int)mod; return a; }
modint operator-=(modint& a, modint b) { a.n -= b.n; if (a.n < 0)a.n += (int)mod; return a; }
modint operator*=(modint& a, modint b) { a.n = ((ll)a.n * b.n) % mod; return a; }
modint operator+(modint a, modint b) { return a += b; }
modint operator-(modint a, modint b) { return a -= b; }
modint operator*(modint a, modint b) { return a *= b; }
modint operator^(modint a, ll n) {
    if (n == 0)return modint(1);
    modint res = (a * a) ^ (n / 2);
    if (n % 2)res = res * a;
    return res;
}

ll inv(ll a, ll p) {
    return (a == 1 ? 1 : (1 - p * inv(p % a, a)) / a + p);
}
modint operator/(modint a, modint b) { return a * modint(inv(b, mod)); }
modint operator/=(modint& a, modint b) { a = a / b; return a; }
const int max_n = 1 << 20;
modint fact[max_n], factinv[max_n];
void init_f() {
    fact[0] = modint(1);
    for (int i = 0; i < max_n - 1; i++) {
        fact[i + 1] = fact[i] * modint(i + 1);
    }
    factinv[max_n - 1] = modint(1) / fact[max_n - 1];
    for (int i = max_n - 2; i >= 0; i--) {
        factinv[i] = factinv[i + 1] * modint(i + 1);
    }
}
modint comb(int a, int b) {
    if (a < 0 || b < 0 || a < b)return 0;
    return fact[a] * factinv[b] * factinv[a - b];
}
modint combP(int a, int b) {
    if (a < 0 || b < 0 || a < b)return 0;
    return fact[a] * factinv[a - b];
}

ll gcd(ll a, ll b) {
    a = abs(a); b = abs(b);
    if (a < b)swap(a, b);
    while (b) {
        ll r = a % b; a = b; b = r;
    }
    return a;
}
template<typename T>
void addv(vector<T>& v, int loc, T val) {
    if (loc >= v.size())v.resize(loc + 1, 0);
    v[loc] += val;
}
/*const int mn = 2000005;
bool isp[mn];
vector<int> ps;
void init() {
    fill(isp + 2, isp + mn, true);
    for (int i = 2; i < mn; i++) {
        if (!isp[i])continue;
        ps.push_back(i);
        for (int j = 2 * i; j < mn; j += i) {
            isp[j] = false;
        }
    }
}*/

//[,val)
template<typename T>
auto prev_itr(set<T>& st, T val) {
    auto res = st.lower_bound(val);
    if (res == st.begin())return st.end();
    res--; return res;
}

//[val,)
template<typename T>
auto next_itr(set<T>& st, T val) {
    auto res = st.lower_bound(val);
    return res;
}
using mP = pair<modint, modint>;
mP operator+(mP a, mP b) {
    return { a.first + b.first,a.second + b.second };
}
mP operator+=(mP& a, mP b) {
    a = a + b; return a;
}
mP operator-(mP a, mP b) {
    return { a.first - b.first,a.second - b.second };
}
mP operator-=(mP& a, mP b) {
    a = a - b; return a;
}
LP operator+(LP a, LP b) {
    return { a.first + b.first,a.second + b.second };
}
LP operator+=(LP& a, LP b) {
    a = a + b; return a;
}
LP operator-(LP a, LP b) {
    return { a.first - b.first,a.second - b.second };
}
LP operator-=(LP& a, LP b) {
    a = a - b; return a;
}

mt19937 mt(time(0));

const string drul = "DRUL";
string senw = "SENW";
//DRUL,or SENW
//int dx[4] = { 1,0,-1,0 };
//int dy[4] = { 0,1,0,-1 };

//------------------------------------


//typedef vector<vector<modint>> mat;
//typedef vector<modint> vec;
//mat mtmul(mat& A, mat& B) {
//    mat C(A.size(), vec(B[0].size()));
//    rep(i, (int)A.size()) {
//        rep(k, (int)B.size()) {
//            rep(j, (int)B[0].size()) {
//                C[i][j] += A[i][k] * B[k][j];
//            }
//        }
//    }
//    return C;
//}
//mat mtpow(mat A, ll n) {
//    mat B(A.size(), vec(A.size()));
//    rep(i, (int)A.size()) {
//        B[i][i] = 1;
//    }
//    while (n > 0) {
//        if (n & (ll)1)B = mtmul(B, A);
//        A = mtmul(A, A);
//        n >>= 1;
//    }
//    return B;
//}
//
//mat mem[30];
//mat A;
//void init() {
//    A = { {4,4},{0,1} };
//}

void vout(int x,int sz=10) {
    while (sz--) {
        cout << x % 2; x /= 2;
    }
    cout << "\n";
}
modint calc0(int k) {
    k--;
    modint val = mod_pow(4, k);
    val -= 1;
    val *= 4;
    val /= 3;
    return val;
}
int calc(string s, int k) {
    int c1 = 0;
    rep(i, s.size())if (s[i] == '1') {
        c1++;
    }
    if (c1 == 0) {
        return calc0(k);
    }
    vector<modint> le(s.size() + 1);
    le[0] = 1;
    rep1(i, s.size()) {
        le[i] = le[i - 1] * (modint)2;
        if (i % 2 == 0)le[i] += 1;
    }
    modint res = 0;
    per(i, s.size()) {
        if (c1 == 0) {
            int sup = (i + 3) / 2;
            if (k <= sup) {
                res += calc0(k);
                break;
            }
            else return -1;
        }
        if (s[i] == '1') {
            //cout << "hello\n";
            res += le[i];
            s[i] = '0';
            c1--;
            if (i % 2) {
                if (s[i - 1] == '0') {
                    c1++; s[i - 1] = '1';
                }
                else{
                    c1--; s[i - 1] = '0';
                }
            }
        }
    }
    return res;
}
string trans(int cur) {
    string res;
    while (cur > 0) {
        res.push_back('0' + (cur % 2)); cur /= 2;
    }
    return res;
}
void expr() {
    set<P> st;
    int cur = 0;
    map<int, int> mp;
    rep(t, 1000) {
        //if (cur == 0)cout << _<<"\n";
        vout(cur); mp[cur]++;
        {
            int c = calc(trans(cur), mp[cur]);
            //cout << trans(cur) << "\n";
            cout << c << "\n";
            assert(c == t);
        }
        bool upd = false;
        rep(i, 20) {
            int to = cur ^ (1 << i);
            if (st.count(minmax(cur, to)))continue;
            st.insert(minmax(cur, to));
            upd = true;
            cur = to;
            break;
        }
        assert(upd);
    }
}
void solve() {
    string s, t; cin >> s >> t;
    int k; cin >> k;
    int sz = max(s.size(), t.size());
    reverse(all(s));
    reverse(all(t));
    s.resize(sz, '0');
    t.resize(sz, '0');
    rep(i, s.size()) {
        if (t[i] == '1') {
            s[i] = '0' + '1' - s[i];
        }
    }
    cout << calc(s, k) << "\n";
}



signed main() {
    ios::sync_with_stdio(false);
    cin.tie(0);
    //cout << fixed<<setprecision(10);
    //init_f();
    //init();
    //while(true)
    //expr();
    int t; cin >> t; rep(i, t)
    solve();
    return 0;
}