Flowers and Chocolate - VietMX's Blog

It’s Piegirl’s birthday soon, and Pieguy has decided to buy her a bouquet of flowers and a basket of chocolates.

The flower shop has F different types of flowers available. The i-th type of flower always has exactly p _i petals. Pieguy has decided to buy a bouquet consisting of exactly N flowers. He may buy the same type of flower multiple times. The N flowers are then arranged into a bouquet. The position of the flowers within a bouquet matters. You can think of a bouquet as an ordered list of flower types.

The chocolate shop sells chocolates in boxes. There are B different types of boxes available. The i-th type of box contains c _i pieces of chocolate. Pieguy can buy any number of boxes, and can buy the same type of box multiple times. He will then place these boxes into a basket. The position of the boxes within the basket matters. You can think of the basket as an ordered list of box types.

Pieguy knows that Piegirl likes to pluck a petal from a flower before eating each piece of chocolate. He would like to ensure that she eats the last piece of chocolate from the last box just after plucking the last petal from the last flower. That is, the total number of petals on all the flowers in the bouquet should equal the total number of pieces of chocolate in all the boxes in the basket.

How many different bouquet+basket combinations can Pieguy buy? The answer may be very large, so compute it modulo 1000000007 = 10⁹ + 7.

Input

The first line of input will contain integers F, B, and N (1 ≤ F ≤ 10, 1 ≤ B ≤ 100, 1 ≤ N ≤ 10¹⁸), the number of types of flowers, the number of types of boxes, and the number of flowers that must go into the bouquet, respectively.

The second line of input will contain F integers p ₁, p ₂, …, p _F (1 ≤ p _i ≤ 10⁹), the numbers of petals on each of the flower types.

The third line of input will contain B integers c ₁, c ₂, …, c _B (1 ≤ c _i ≤ 250), the number of pieces of chocolate in each of the box types.

Output

Print the number of bouquet+basket combinations Pieguy can buy, modulo 1000000007 = 10⁹ + 7.

Examples

input

2 3 3
3 5
10 3 7

output

input

6 5 10
9 3 3 4 9 9
9 9 1 6 4

output

31415926

Note

In the first example, there is 1 way to make a bouquet with 9 petals (3 + 3 + 3), and 1 way to make a basket with 9 pieces of chocolate (3 + 3 + 3), for 1 possible combination. There are 3 ways to make a bouquet with 13 petals (3 + 5 + 5, 5 + 3 + 5, 5 + 5 + 3), and 5 ways to make a basket with 13 pieces of chocolate (3 + 10, 10 + 3, 3 + 3 + 7, 3 + 7 + 3, 7 + 3 + 3), for 15 more combinations. Finally there is 1 way to make a bouquet with 15 petals (5 + 5 + 5) and 1 way to make a basket with 15 pieces of chocolate (3 + 3 + 3 + 3 + 3), for 1 more combination.

Note that it is possible for multiple types of flowers to have the same number of petals. Such types are still considered different. Similarly different types of boxes may contain the same number of pieces of chocolate, but are still considered different.

#include <bits/stdc++.h>
 
using namespace std;
 
string to_string(string s) {
return '"' + s + '"';
}
 
string to_string(const char* s) {
return to_string((string) s);
}
 
string to_string(bool b) {
return (b ? "true" : "false");
}
 
template <typename A, typename B>
string to_string(pair<A, B> p) {
return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}
 
template <typename A>
string to_string(A v) {
bool first = true;
string res = "{";
for (const auto &x : v) {
if (!first) {
res += ", ";
}
first = false;
res += to_string(x);
}
res += "}";
return res;
}
 
void debug_out() { cerr << endl; }
 
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << to_string(H);
  debug_out(T...);
}
 
#ifdef LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif
 
template <typename T>
class Modular {
public:
using Type = typename decay<decltype(T::value)>::type;
 
constexpr Modular() : value() {}
template <typename U>
Modular(const U& x) {
value = normalize(x);
}
 
template <typename U>
static Type normalize(const U& x) {
Type v;
if (-mod() <= x && x < mod()) v = static_cast<Type>(x);
else v = static_cast<Type>(x % mod());
if (v < 0) v += mod();
    return v;
  }
 
  const Type& operator()() const { return value; }
  template <typename U>
explicit operator U() const { return static_cast<U>(value); }
constexpr static Type mod() { return T::value; }
 
Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; }
Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; }
  template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); }
template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); }
Modular& operator++() { return *this += 1; }
Modular& operator--() { return *this -= 1; }
Modular operator++(int) { Modular result(*this); *this += 1; return result; }
Modular operator--(int) { Modular result(*this); *this -= 1; return result; }
Modular operator-() const { return Modular(-value); }
 
template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) {
value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
return *this;
}
template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) {
int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod());
value = normalize(value * rhs.value - q * mod());
return *this;
}
template <typename U = T>
typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) {
value = normalize(value * rhs.value);
return *this;
}
 
Modular inverse() const {
Type a = value, b = mod(), u = 0, v = 1;
while (a != 0) {
Type t = b / a;
b -= t * a; swap(a, b);
u -= t * v; swap(u, v);
}
assert(b == 1);
return Modular(u);
}
Modular& operator/=(const Modular& other) { return *this *= other.inverse(); }
 
template <typename U>
friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs);
 
template <typename U>
friend std::istream& operator>>(std::istream& stream, Modular<U>& number);
 
private:
Type value;
};
 
template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; }
template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); }
template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; }
 
template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
 
template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
 
template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
 
template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
 
template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
 
template<typename T, typename U>
Modular<T> power(const Modular<T>& a, const U& b) {
assert(b >= 0);
Modular<T> x = a, res = 1;
U p = b;
while (p > 0) {
if (p & 1) res *= x;
x *= x;
p >>= 1;
}
return res;
}
 
template <typename T>
string to_string(const Modular<T>& number) {
return to_string(number());
}
 
template <typename T>
std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) {
return stream << number();
}
 
template <typename T>
std::istream& operator>>(std::istream& stream, Modular<T>& number) {
stream >> number.value;
number.value = Modular<T>::normalize(number.value);
return stream;
}
 
struct VarMod { static int value; };
int VarMod::value;
 
//using Mint = Modular<VarMod>;
using Mint = Modular<std::integral_constant<int, (int) 1e9 + 7>>;
 
template <typename T>
vector<T>& operator+=(vector<T>& a, const vector<T>& b) {
if (a.size() < b.size()) {
    a.resize(b.size());
  }
  for (int i = 0; i < (int) b.size(); i++) {
    a[i] += b[i];
  }
  return a;
}
 
template <typename T>
vector<T> operator+(const vector<T>& a, const vector<T>& b) {
vector<T> c = a;
return c += b;
}
 
template <typename T>
vector<T>& operator-=(vector<T>& a, const vector<T>& b) {
if (a.size() < b.size()) {
    a.resize(b.size());
  }
  for (int i = 0; i < (int) b.size(); i++) {
    a[i] -= b[i];
  }
  return a;
}
 
template <typename T>
vector<T> operator-(const vector<T>& a, const vector<T>& b) {
vector<T> c = a;
return c -= b;
}
 
template <typename T>
vector<T>& operator*=(vector<T>& a, const vector<T>& b) {
if (a.empty() || b.empty()) {
a.clear();
} else {
vector<T> c = a;
a.assign(a.size() + b.size() - 1, 0);
for (int i = 0; i < (int) c.size(); i++) {
      for (int j = 0; j < (int) b.size(); j++) {
        a[i + j] += c[i] * b[j];
      }
    }
  }
  return a;
}
 
template <typename T>
vector<T> operator*(const vector<T>& a, const vector<T>& b) {
vector<T> c = a;
return c *= b;
}
 
template <typename T>
vector<T> inverse(const vector<T>& a) {
assert(!a.empty());
int n = (int) a.size();
vector<T> b = {1 / a[0]};
while ((int) b.size() < n) {
    vector<T> x = b * b * vector<T>(a.begin(), a.begin() + min(a.size(), b.size() << 1));
    b.resize(b.size() << 1);
    for (int i = (int) b.size() >> 1; i < (int) min(x.size(), b.size()); i++) {
      b[i] = -x[i];
    }
  }
  b.resize(n);
  return b;
}
 
template <typename T>
vector<T>& operator/=(vector<T>& a, const vector<T>& b) {
int n = (int) a.size();
int m = (int) b.size();
if (n < m) {
    a.clear();
  } else {
    vector<T> d = b;
reverse(a.begin(), a.end());
reverse(d.begin(), d.end());
d.resize(n - m + 1);
a *= inverse(d);
a.erase(a.begin() + n - m + 1, a.end());
reverse(a.begin(), a.end());
}
return a;
}
 
template <typename T>
vector<T> operator/(const vector<T>& a, const vector<T>& b) {
vector<T> c = a;
return c /= b;
}
 
template <typename T>
vector<T>& operator%=(vector<T>& a, const vector<T>& b) {
int n = (int) a.size();
int m = (int) b.size();
if (n >= m) {
vector<T> c = (a / b) * b;
a.resize(m - 1);
for (int i = 0; i < m - 1; i++) {
      a[i] -= c[i];
    }
  }
  return a;
}
 
template <typename T>
vector<T> operator%(const vector<T>& a, const vector<T>& b) {
vector<T> c = a;
return c %= b;
}
 
template <typename T, typename U>
vector<T> power(const vector<T>& a, const U& b, const vector<T>& c) {
assert(b >= 0);
vector<U> binary;
U bb = b;
while (bb > 0) {
binary.push_back(bb & 1);
bb >>= 1;
}
vector<T> res = vector<T> {1} % c;
for (int j = (int) binary.size() - 1; j >= 0; j--) {
res = res * res % c;
if (binary[j] == 1) {
res = res * a % c;
}
}
return res;
}
 
namespace fft {
 
typedef double dbl;
 
struct num {
dbl x, y;
num() { x = y = 0; }
num(dbl x_, dbl y_) : x(x_), y(y_) {}
};
 
inline num operator+(num a, num b) { return num(a.x + b.x, a.y + b.y); }
inline num operator-(num a, num b) { return num(a.x - b.x, a.y - b.y); }
inline num operator*(num a, num b) { return num(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x); }
inline num conj(num a) { return num(a.x, -a.y); }
 
int base = 1;
vector<num> roots = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
 
const dbl PI = static_cast<dbl>(acosl(-1.0));
 
void ensure_base(int nbase) {
if (nbase <= base) {
    return;
  }
  rev.resize(1 << nbase);
  for (int i = 0; i < (1 << nbase); i++) {
    rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
  }
  roots.resize(1 << nbase);
  while (base < nbase) {
    dbl angle = 2 * PI / (1 << (base + 1));
//      num z(cos(angle), sin(angle));
    for (int i = 1 << (base - 1); i < (1 << base); i++) {
      roots[i << 1] = roots[i];
//        roots[(i << 1) + 1] = roots[i] * z;
      dbl angle_i = angle * (2 * i + 1 - (1 << base));
      roots[(i << 1) + 1] = num(cos(angle_i), sin(angle_i));
    }
    base++;
  }
}
 
void fft(vector<num> &a, int n = -1) {
if (n == -1) {
n = (int) a.size();
}
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
    if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
    for (int i = 0; i < n; i += 2 * k) {
      for (int j = 0; j < k; j++) {
        num z = a[i + j + k] * roots[j + k];
        a[i + j + k] = a[i + j] - z;
        a[i + j] = a[i + j] + z;
      }
    }
  }
}
 
vector<num> fa, fb;
 
vector<int64_t> square(const vector<int>& a) {
if (a.empty()) {
return {};
}
int need = (int) a.size() + (int) a.size() - 1;
int nbase = 1;
while ((1 << nbase) < need) nbase++;
  ensure_base(nbase);
  int sz = 1 << nbase;
  if ((sz >> 1) > (int) fa.size()) {
fa.resize(sz >> 1);
}
for (int i = 0; i < (sz >> 1); i++) {
int x = (2 * i < (int) a.size() ? a[2 * i] : 0);
    int y = (2 * i + 1 < (int) a.size() ? a[2 * i + 1] : 0);
    fa[i] = num(x, y);
  }
  fft(fa, sz >> 1);
num r(1.0 / (sz >> 1), 0.0);
for (int i = 0; i <= (sz >> 2); i++) {
int j = ((sz >> 1) - i) & ((sz >> 1) - 1);
num fe = (fa[i] + conj(fa[j])) * num(0.5, 0);
num fo = (fa[i] - conj(fa[j])) * num(0, -0.5);
num aux = fe * fe + fo * fo * roots[(sz >> 1) + i] * roots[(sz >> 1) + i];
num tmp = fe * fo;
fa[i] = r * (conj(aux) + num(0, 2) * conj(tmp));
fa[j] = r * (aux + num(0, 2) * tmp);
}
fft(fa, sz >> 1);
vector<int64_t> res(need);
for (int i = 0; i < need; i++) {
    res[i] = llround(i % 2 == 0 ? fa[i >> 1].x : fa[i >> 1].y);
}
return res;
}
 
vector<int64_t> multiply(const vector<int>& a, const vector<int>& b) {
if (a.empty() || b.empty()) {
return {};
}
if (a.size() == b.size() && a == b) {
return square(a);
}
int need = (int) a.size() + (int) b.size() - 1;
int nbase = 1;
while ((1 << nbase) < need) nbase++;
  ensure_base(nbase);
  int sz = 1 << nbase;
  if (sz > (int) fa.size()) {
fa.resize(sz);
}
for (int i = 0; i < sz; i++) {
    int x = (i < (int) a.size() ? a[i] : 0);
    int y = (i < (int) b.size() ? b[i] : 0);
    fa[i] = num(x, y);
  }
  fft(fa, sz);
  num r(0, -0.25 / (sz >> 1));
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
num z = (fa[j] * fa[j] - conj(fa[i] * fa[i])) * r;
fa[j] = (fa[i] * fa[i] - conj(fa[j] * fa[j])) * r;
fa[i] = z;
}
for (int i = 0; i < (sz >> 1); i++) {
num A0 = (fa[i] + fa[i + (sz >> 1)]) * num(0.5, 0);
num A1 = (fa[i] - fa[i + (sz >> 1)]) * num(0.5, 0) * roots[(sz >> 1) + i];
fa[i] = A0 + A1 * num(0, 1);
}
fft(fa, sz >> 1);
vector<int64_t> res(need);
for (int i = 0; i < need; i++) {
    res[i] = llround(i % 2 == 0 ? fa[i >> 1].x : fa[i >> 1].y);
}
return res;
}
 
vector<int> multiply_mod(const vector<int>& a, const vector<int>& b, int m) {
if (a.empty() || b.empty()) {
return {};
}
int eq = (a.size() == b.size() && a == b);
int need = (int) a.size() + (int) b.size() - 1;
int nbase = 0;
while ((1 << nbase) < need) nbase++;
  ensure_base(nbase);
  int sz = 1 << nbase;
  if (sz > (int) fa.size()) {
fa.resize(sz);
}
for (int i = 0; i < (int) a.size(); i++) {
    int x = (a[i] % m + m) % m;
    fa[i] = num(x & ((1 << 15) - 1), x >> 15);
}
fill(fa.begin() + a.size(), fa.begin() + sz, num {0, 0});
fft(fa, sz);
if (sz > (int) fb.size()) {
fb.resize(sz);
}
if (eq) {
copy(fa.begin(), fa.begin() + sz, fb.begin());
} else {
for (int i = 0; i < (int) b.size(); i++) {
      int x = (b[i] % m + m) % m;
      fb[i] = num(x & ((1 << 15) - 1), x >> 15);
}
fill(fb.begin() + b.size(), fb.begin() + sz, num {0, 0});
fft(fb, sz);
}
dbl ratio = 0.25 / sz;
num r2(0, -1);
num r3(ratio, 0);
num r4(0, -ratio);
num r5(0, 1);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
num a1 = (fa[i] + conj(fa[j]));
num a2 = (fa[i] - conj(fa[j])) * r2;
num b1 = (fb[i] + conj(fb[j])) * r3;
num b2 = (fb[i] - conj(fb[j])) * r4;
if (i != j) {
num c1 = (fa[j] + conj(fa[i]));
num c2 = (fa[j] - conj(fa[i])) * r2;
num d1 = (fb[j] + conj(fb[i])) * r3;
num d2 = (fb[j] - conj(fb[i])) * r4;
fa[i] = c1 * d1 + c2 * d2 * r5;
fb[i] = c1 * d2 + c2 * d1;
}
fa[j] = a1 * b1 + a2 * b2 * r5;
fb[j] = a1 * b2 + a2 * b1;
}
fft(fa, sz);
fft(fb, sz);
vector<int> res(need);
for (int i = 0; i < need; i++) {
    int64_t aa = llround(fa[i].x);
    int64_t bb = llround(fb[i].x);
    int64_t cc = llround(fa[i].y);
    res[i] = static_cast<int>((aa + ((bb % m) << 15) + ((cc % m) << 30)) % m);
  }
  return res;
}
 
}  // namespace fft
 
template <typename T>
typename enable_if<is_same<typename Modular<T>::Type, int>::value, vector<Modular<T>>>::type& operator*=(
vector<Modular<T>>& a,
const vector<Modular<T>>& b) {
vector<int> A(a.size());
for (int i = 0; i < (int) A.size(); i++) {
    A[i] = static_cast<int>(a[i]);
}
vector<int> B(b.size());
for (int i = 0; i < (int) B.size(); i++) {
    B[i] = static_cast<int>(b[i]);
}
vector<int> C = fft::multiply_mod(A, B, T::value);
a.resize(C.size());
for (int i = 0; i < (int) C.size(); i++) {
    a[i] = C[i];
  }
  return a;
}
 
template <typename T>
typename enable_if<is_same<typename Modular<T>::Type, int>::value, vector<Modular<T>>>::type operator*(
const vector<Modular<T>>& a,
const vector<Modular<T>>& b) {
vector<Modular<T>> c = a;
return c *= b;
}
 
int main() {
ios::sync_with_stdio(false);
cin.tie(0);
int f, b;
long long n;
cin >> f >> b >> n;
vector<int> p(f);
for (int i = 0; i < f; i++) {
    cin >> p[i];
}
vector<int> c(b);
for (int i = 0; i < b; i++) {
    cin >> c[i];
}
int mx = *max_element(c.begin(), c.end());
vector<Mint> md(mx + 1);
md[mx]++;
for (int i = 0; i < b; i++) {
    md[mx - c[i]]--;
  }
  vector<Mint> sum;
for (int i = 0; i < f; i++) {
    sum += power({0, 1}, p[i], md);
  }
  vector<Mint> res = power(sum, n, md);
vector<Mint> final(mx);
final[mx - 1] = 1;
res = res * final % md;
debug(res);
cout << ((int) res.size() <= mx - 1 ? Mint(0) : res[mx - 1]) << '\n';
  return 0;
}