#line 1 "main.cpp"
#include <bits/stdc++.h>
// #include <ext/pb_ds/assoc_container.hpp>
// #include <ext/pb_ds/tag_and_trait.hpp>
// #include <ext/pb_ds/tree_policy.hpp>
// namespace gt = __gnu_pbds;
#define IS_MULTITEST 1

using namespace std;

#line 2 "/Library/angel/math/modint.hpp"

#line 2 "/Library/Nyaan/modint/montgomery-modint.hpp"



template <uint32_t mod>
struct LazyMontgomeryModInt {
  using mint = LazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }

  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
  static_assert(r * mod == 1, "invalid, r * mod != 1");
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");

  u32 a;

  constexpr LazyMontgomeryModInt() : a(0) {}
  constexpr LazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};

  static constexpr u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }

  constexpr mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  constexpr mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
  constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
  constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
  constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
  constexpr bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr mint operator-() const { return mint() - mint(*this); }

  constexpr mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  
  constexpr mint inverse() const { return pow(mod - 2); }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = LazyMontgomeryModInt<mod>(t);
    return (is);
  }
  
  constexpr u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static constexpr u32 get_mod() { return mod; }
};
#line 2 "/Library/Nyaan/modulo/binomial.hpp"

template <typename T>
struct Binomial {
  vector<T> f, g, h;
  Binomial(int MAX = 0) {
    assert(T::get_mod() != 0 && "Binomial<mint>()");
    f.resize(1, T{1});
    g.resize(1, T{1});
    h.resize(1, T{1});
    while (MAX >= (int)f.size()) extend();
  }

  void extend() {
    int n = f.size();
    int m = n * 2;
    f.resize(m);
    g.resize(m);
    h.resize(m);
    for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
    g[m - 1] = f[m - 1].inverse();
    h[m - 1] = g[m - 1] * f[m - 2];
    for (int i = m - 2; i >= n; i--) {
      g[i] = g[i + 1] * T(i + 1);
      h[i] = g[i] * f[i - 1];
    }
  }

  T fac(int i) {
    if (i < 0) return T(0);
    while (i >= (int)f.size()) extend();
    return f[i];
  }

  T finv(int i) {
    if (i < 0) return T(0);
    while (i >= (int)g.size()) extend();
    return g[i];
  }

  T inv(int i) {
    if (i < 0) return -inv(-i);
    while (i >= (int)h.size()) extend();
    return h[i];
  }

  T C(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r) * finv(r);
  }

  inline T operator()(int n, int r) { return C(n, r); }

  template <typename I>
  T multinomial(const vector<I>& r) {
    static_assert(is_integral<I>::value == true);
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return T(0);
      n += x;
    }
    T res = fac(n);
    for (auto& x : r) res *= finv(x);
    return res;
  }

  template <typename I>
  T operator()(const vector<I>& r) {
    return multinomial(r);
  }

  T C_naive(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    T ret = T(1);
    r = min(r, n - r);
    for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
    return ret;
  }

  T P(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r);
  }

  // [x^r] 1 / (1-x)^n
  T H(int n, int r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};
#line 5 "/Library/angel/math/modint.hpp"

using Modint1000000007 = LazyMontgomeryModInt<1000000007>;
using Binomial1000000007 = Binomial<Modint1000000007>;
using Modint998244353 = LazyMontgomeryModInt<998244353>;
using Binomial998244353 = Binomial<Modint998244353>;
#line 11 "main.cpp"

#pragma region Macros
// clang-format off
using ll = long long; using uint = unsigned int; using ull = unsigned long long;
using i32 = int; using i64 = ll; using u32 = uint; using u64 = ull;
using i128 = __int128_t; using u128 = __uint128_t;
using Str = string;
template <class T> using Vec = vector<T>;
template <class T> using RevPriq = priority_queue<T, vector<T>, greater<T>>;
constexpr std::array<std::pair<int, int>, 4> dxy4 = {{{-1, 0}, {0, 1}, {1, 0}, {0, -1}}};
constexpr std::array<std::pair<int, int>, 8> dxy8 = {
    {{-1, -1}, {-1, 0}, {-1, 1}, {0, 1}, {1, 1}, {1, 0}, {1, -1}, {0, -1}}};
constexpr int inf32 = 1 << 30; constexpr ll inf64 = 1ll << 60;
constexpr char eoln = '\n';
#define L(i, l, r) for (int i = (l); i < (r); ++i)
#define R(i, l, r) for (int i = (r) - 1; i >= (l); --i)
#define ALL(x) (x).begin(), (x).end()
#define mem(a, x) memset((a), (x), sizeof(a))
#define sz(a) (int)((a).size())
#define fi first
#define se second
#define pb push_back
#define mp make_pair
// clang-format on
#pragma endregion

// Coding Space

// vars

// funcs

using Fp = Modint1000000007;

Fp ml(ll n) {
    return (Fp(4) / Fp(3)) * Fp(4).pow(n - 1) - Fp(4) / Fp(3);
}

void main_() {
    Str S, T;
    cin >> S >> T;
    int k;
    cin >> k;
    if (sz(S) < sz(T)) S = Str(sz(T) - sz(S), '0') + S;
    else T = Str(sz(S) - sz(T), '0') + T;
    int N = sz(S);
    L(i, 0, N) {
        if (S[i] == T[i]) S[i] = T[i] = '0';
        else S[i] = '0', T[i] = '1';
    }
    reverse(ALL(S));
    reverse(ALL(T));
    while (not S.empty()) {
        if (S.back() == '0' and T.back() == '0') {
            S.pop_back();
            T.pop_back();
        } else {
            break;
        }
    }
    reverse(ALL(S));
    reverse(ALL(T));
    N = sz(S);
    if (N == 0) {
        cout << ml(k) << endl;
    } else {
        int c = 0, f = 0;
        R(i, 0, N) {
            if (T[i] == '0') ++c;
            else {
                f = i;
                break;
            }
        }
        if (k > c / 2 + 1) {
            cout << -1 << endl;
            return;
        }

        cerr << S << ' ' << T << endl;
        Fp base = 0;
        if (N % 2 == 1) {
            S = '0' + S;
            T = '0' + T;
            ++f;
            ++N;
        }
        for (int i = 0; i < N; i += 2) {
            Fp v = ml(N / 2 - i / 2);
            if (T[i] == '0' and T[i + 1] == '1') base += v + 1;
            else if (T[i] == '1' and T[i + 1] == '1') base += Fp(2) * v + 2;
            else if (T[i] == '1') base += Fp(3) * v + 3;
        }
        Fp iad = 0;
        L(i, 0, k - 1) {
            iad += ml(i + 2);
        }
        cout << base + iad << endl;
    }
}

set<pair<int, int>> s;
map<int, int> c;

void main_2() {
    int n = 0;
    int test = 100000;
    while (test--) {
        ++c[n];
        L(i, 0, 30) {
            if (s.find(minmax(n, n xor (1 << i))) == s.end()) {
                s.insert(minmax(n, n xor (1 << i)));
                n = n xor (1 << i);
                break;
            }
        }
    }
    for (auto [a, b] : c) cout << bitset<11>(a) << ' ' << b << endl;
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    if constexpr (IS_MULTITEST == 0) {
        main_2();
        // main_();
    } else {
        // multitest (cf-style)
        int T;
        cin >> T;
        while (T--) {
            main_();
            cout << flush;
        }
    }
}