#include using namespace std; using int64 = long long; const int mod = 1e9 + 7; // const int mod = 998244353; const int64 infll = (1LL << 62) - 1; const int inf = (1 << 30) - 1; struct IoSetup { IoSetup() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(10); cerr << fixed << setprecision(10); } } iosetup; template< typename T1, typename T2 > ostream &operator<<(ostream &os, const pair< T1, T2 > &p) { os << p.first << " " << p.second; return os; } template< typename T1, typename T2 > istream &operator>>(istream &is, pair< T1, T2 > &p) { is >> p.first >> p.second; return is; } template< typename T > ostream &operator<<(ostream &os, const vector< T > &v) { for(int i = 0; i < (int) v.size(); i++) { os << v[i] << (i + 1 != v.size() ? " " : ""); } return os; } template< typename T > istream &operator>>(istream &is, vector< T > &v) { for(T &in : v) is >> in; return is; } template< typename T1, typename T2 > inline bool chmax(T1 &a, T2 b) { return a < b && (a = b, true); } template< typename T1, typename T2 > inline bool chmin(T1 &a, T2 b) { return a > b && (a = b, true); } template< typename T = int64 > vector< T > make_v(size_t a) { return vector< T >(a); } template< typename T, typename... Ts > auto make_v(size_t a, Ts... ts) { return vector< decltype(make_v< T >(ts...)) >(a, make_v< T >(ts...)); } template< typename T, typename V > typename enable_if< is_class< T >::value == 0 >::type fill_v(T &t, const V &v) { t = v; } template< typename T, typename V > typename enable_if< is_class< T >::value != 0 >::type fill_v(T &t, const V &v) { for(auto &e : t) fill_v(e, v); } template< typename F > struct FixPoint : F { FixPoint(F &&f) : F(forward< F >(f)) {} template< typename... Args > decltype(auto) operator()(Args &&... args) const { return F::operator()(*this, forward< Args >(args)...); } }; template< typename F > inline decltype(auto) MFP(F &&f) { return FixPoint< F >{forward< F >(f)}; } template< typename T > T mod_pow(T x, T n, const T &p) { T ret = 1; while(n > 0) { if(n & 1) (ret *= x) %= p; (x *= x) %= p; n >>= 1; } return ret; } static constexpr uint32_t get_r(int mod) { uint64_t ret = 1, m = mod, n = mod - 2; while(n) { ret = uint32_t(ret * m); m = uint32_t(m * m); n >>= 1; } return ret; }; template< uint32_t mod > struct LazyMontgomeryModInt { using mint = LazyMontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 r = get_r(mod); 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; LazyMontgomeryModInt() : a(0) {} LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)) {}; static u32 reduce(const u64 &b) { return u32(b >> 32) + mod - u32((u64(u32(b) * r) * mod) >> 32); } mint &operator+=(const mint &b) { if(i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } mint &operator-=(const mint &b) { if(i32(a -= b.a) < 0) a += 2 * mod; return *this; } mint &operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } mint &operator/=(const mint &b) { *this *= b.inverse(); return *this; } mint operator+(const mint &b) const { return mint(*this) += b; } mint operator-(const mint &b) const { return mint(*this) -= b; } mint operator*(const mint &b) const { return mint(*this) *= b; } mint operator/(const mint &b) const { return mint(*this) /= b; } u32 get() const { u32 ret = reduce(a); return ret >= mod ? ret - mod : ret; } mint pow(u64 n) const { mint ret(1), mul(*this); while(n > 0) { if(n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } 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); } mint inverse() const { return pow(mod - 2); } static constexpr u32 get_mod() { return mod; } }; static constexpr uint32_t get_pr(uint32_t mod) { using u64 = uint64_t; u64 ds[32] = {}; int idx = 0; u64 m = mod - 1; for(u64 i = 2; i * i <= m; ++i) { if(m % i == 0) { ds[idx++] = i; while(m % i == 0) m /= i; } } if(m != 1) ds[idx++] = m; uint32_t pr = 2; while(1) { int flg = 1; for(int i = 0; i < idx; i++) { u64 a = pr, b = (mod - 1) / ds[i], r = 1; while(b) { if(b & 1) r = r * a % mod; a = a * a % mod; b >>= 1; } if(r == 1) { flg = 0; break; } } if(flg == 1) break; ++pr; } return pr; }; template< typename mint > struct NTT { static constexpr uint32_t mod = mint::get_mod(); static constexpr uint32_t pr = get_pr(mod); static constexpr int level = __builtin_ctzll(mod - 1); mint dw[level], dy[level]; void setwy(int k) { mint w[level], y[level]; w[k - 1] = mint(pr).pow((mod - 1) / (1 << k)); y[k - 1] = w[k - 1].inverse(); for(int i = k - 2; i > 0; --i) w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1]; dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2]; for(int i = 3; i < k; ++i) { dw[i] = dw[i - 1] * y[i - 2] * w[i]; dy[i] = dy[i - 1] * w[i - 2] * y[i]; } } void fft4(vector< mint > &a, int k) { if(k & 1) { int v = 1 << (k - 1); for(int j = 0; j < v; ++j) { mint ajv = a[j + v]; a[j + v] = a[j] - ajv; a[j] += ajv; } } int u = 1 << (2 + (k & 1)); int v = 1 << (k - 2 - (k & 1)); mint one = mint(1); mint imag = dw[1]; while(v) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = j1 + v; int j3 = j2 + v; for(; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3; } } // jh >= 1 mint ww = one, xx = one * dw[2], wx = one; for(int jh = 4; jh < u;) { ww = xx * xx, wx = ww * xx; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for(; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3; } xx *= dw[__builtin_ctzll((jh += 4))]; } u <<= 2; v >>= 2; } } void ifft4(vector< mint > &a, int k) { int u = 1 << (k - 2); int v = 1; mint one = mint(1); mint imag = dy[1]; while(u) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = v + v; int j3 = j2 + v; for(; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag; a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3; a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3; } } // jh >= 1 mint ww = one, xx = one * dy[2], yy = one; u <<= 2; for(int jh = 4; jh < u;) { ww = xx * xx, yy = xx * imag; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for(; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy; a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww; a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww; } xx *= dy[__builtin_ctzll(jh += 4)]; } u >>= 4; v <<= 2; } if(k & 1) { u = 1 << (k - 1); for(int j = 0; j < u; ++j) { mint ajv = a[j] - a[j + u]; a[j] += a[j + u]; a[j + u] = ajv; } } } vector< mint > multiply(const vector< mint > &a, const vector< mint > &b) { int l = a.size() + b.size() - 1; int k = 2, M = 4; while(M < l) M <<= 1, ++k; #ifdef NyaanDebug assert(k <= level); #endif setwy(k); vector< mint > s(M), t(M); for(int i = 0; i < (int) a.size(); ++i) s[i] = a[i]; for(int i = 0; i < (int) b.size(); ++i) t[i] = b[i]; fft4(s, k); fft4(t, k); for(int i = 0; i < M; ++i) s[i] *= t[i]; ifft4(s, k); s.resize(l); mint invm = mint(M).inverse(); for(int i = 0; i < l; ++i) s[i] *= invm; return s; } }; // LazyMontgomeryModInt専用。2^24まで可能 namespace ArbitaryNTT { constexpr int32_t m0 = 167772161; constexpr int32_t m1 = 469762049; constexpr int32_t m2 = 754974721; using mint0 = LazyMontgomeryModInt< m0 >; using mint1 = LazyMontgomeryModInt< m1 >; using mint2 = LazyMontgomeryModInt< m2 >; // modint用 template< typename submint, typename mint > vector< submint > mul(const vector< mint > &a, const vector< mint > &b) { int l = a.size() + b.size() - 1; int k = 2, M = 4; while(M < l) M <<= 1, ++k; NTT< submint > ntt; vector< submint > s(M), t(M); for(int i = 0; i < (int) a.size(); ++i) { s[i].a = submint::reduce(uint64_t(a[i].a) * submint::n2); } for(int i = 0; i < (int) b.size(); ++i) { t[i].a = submint::reduce(uint64_t(b[i].a) * submint::n2); } ntt.setwy(k); ntt.fft4(s, k); ntt.fft4(t, k); for(int i = 0; i < M; ++i) s[i] *= t[i]; ntt.ifft4(s, k); s.resize(l); submint invm = submint(M).inverse(); for(int i = 0; i < l; ++i) s[i] *= invm; return s; } template< typename mint > vector< mint > multiply(const vector< mint > &s, const vector< mint > &t) { auto d0 = mul< mint0 >(s, t); auto d1 = mul< mint1 >(s, t); auto d2 = mul< mint2 >(s, t); int n = d0.size(); vector< mint > res(n); using i64 = int64_t; static const int r01 = mint1(m0).inverse().get(); static const int r02 = mint2(m0).inverse().get(); static const int r12 = mint2(m1).inverse().get(); static const int r02r12 = i64(r02) * r12 % m2; static const int w1 = m0 % mint::get_mod(); static const int w2 = i64(w1) * m1 % mint::get_mod(); // reduceに代入出来る数はmod * 2^32未満。一方、 // a + b * w1 + c * w2 // <= (m0-1) + (m1-1)*(mod-1) + (m2-1)*(mod-1) // = (m0-1) + (m1+m2-2)*(mod-1) // < 2^32+2^32*(mod-1) = 2^32*mod // が導かれるので、for文内部の5行目でmodを取らなくて良いとわかる。 for(int i = 0; i < n; i++) { i64 n1 = d1[i].get(), n2 = d2[i].get(); i64 a = d0[i].get(); i64 b = (n1 + m1 - a) * r01 % m1; i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2; res[i].a = mint::reduce(a + b * w1 + c * w2); } return res; } // int用 template< typename submint, int mod > vector< submint > int_friendly_mul(const vector< int > &a, const vector< int > &b) { int l = a.size() + b.size() - 1; int k = 2, M = 4; while(M < l) M <<= 1, ++k; NTT< submint > ntt; vector< submint > s(M), t(M); for(int i = 0; i < (int) a.size(); ++i) { s[i].a = submint::reduce(uint64_t(a[i]) * submint::n2); } for(int i = 0; i < (int) b.size(); ++i) { t[i].a = submint::reduce(uint64_t(b[i]) * submint::n2); } ntt.setwy(k); ntt.fft4(s, k); ntt.fft4(t, k); for(int i = 0; i < M; ++i) s[i] = s[i] * t[i]; ntt.ifft4(s, k); s.resize(l); submint invm = submint(M).inverse(); for(int i = 0; i < l; ++i) s[i] *= invm; return s; } template< int mod > vector< int > int_friendly_multiply(const vector< int > &s, const vector< int > &t) { auto d0 = int_friendly_mul< mint0, mod >(s, t); auto d1 = int_friendly_mul< mint1, mod >(s, t); auto d2 = int_friendly_mul< mint2, mod >(s, t); int n = d0.size(); vector< int > res(n); using i64 = int64_t; static const int r01 = mint1(m0).inverse().get(); static const int r02 = mint2(m0).inverse().get(); static const int r12 = mint2(m1).inverse().get(); static const int r02r12 = i64(r02) * r12 % m2; static const int w1 = m0 % mod; static const int w2 = i64(w1) * m1 % mod; // reduceに代入出来る数はmod * 2^32未満。一方、 // a + b * w1 + c * w2 // <= (m0-1) + (m1-1)*(mod-1) + (m2-1)*(mod-1) // = (m0-1) + (m1+m2-2)*(mod-1) // < 2^32+2^32*(mod-1) = 2^32*mod // が導かれるので、for文内部の5行目でmodを取らなくて良いとわかる。 for(int i = 0; i < n; i++) { i64 n1 = d1[i].get(), n2 = d2[i].get(); i64 a = d0[i].get(); i64 b = (n1 + m1 - a) * r01 % m1; i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2; res[i] = (a + b * w1 + c * w2) % mod; } return res; } } int main() { int P, N, K, B; cin >> P >> N >> K >> B; vector< int64 > A(N); cin >> A; vector< int64 > beet(P); for(int i = 0; i < P; i++) { beet[mod_pow< int64 >(i, K, P)]++; } constexpr uint32_t MOD = 1000000007; using mint = LazyMontgomeryModInt< MOD >; auto dp = make_v< mint >(P); dp[0] = 1; for(int i = 0; i < N; i++) { vector< int > shift(P); for(int k = 0; k < P; k++) shift[k] = 1LL * A[i] * k % P; vector< mint > dp3(P); for(int k = 0; k < P; k++) dp3[shift[k]] = beet[k]; auto dp2 = ArbitaryNTT::multiply(dp, dp3); for(int k = 0; k < dp.size(); k++) dp[k] = 0; for(int k = 0; k < dp2.size(); k++) dp[k % P] += dp2[k]; } cout << dp[B].get() << endl; }