#include using namespace std; #define rep(i, n) for (int i = 0; i < n; i++) #define rep2(i, x, n) for (int i = x; i <= n; i++) #define rep3(i, x, n) for (int i = x; i >= n; i--) #define each(e, v) for (auto &e : v) #define pb push_back #define eb emplace_back #define all(x) x.begin(), x.end() #define rall(x) x.rbegin(), x.rend() #define sz(x) (int)x.size() using ll = long long; using pii = pair; using pil = pair; using pli = pair; using pll = pair; template bool chmax(T &x, const T &y) { return (x < y) ? (x = y, true) : false; } template bool chmin(T &x, const T &y) { return (x > y) ? (x = y, true) : false; } template int flg(T x, int i) { return (x >> i) & 1; } template void print(const vector &v, T x = 0) { int n = v.size(); for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' '); } template void printn(const vector &v, T x = 0) { int n = v.size(); for (int i = 0; i < n; i++) cout << v[i] + x << '\n'; } template int lb(const vector &v, T x) { return lower_bound(begin(v), end(v), x) - begin(v); } template int ub(const vector &v, T x) { return upper_bound(begin(v), end(v), x) - begin(v); } template void rearrange(vector &v) { sort(begin(v), end(v)); v.erase(unique(begin(v), end(v)), end(v)); } template vector id_sort(const vector &v, bool greater = false) { int n = v.size(); vector ret(n); iota(begin(ret), end(ret), 0); sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; }); return ret; } struct io_setup { io_setup() { ios_base::sync_with_stdio(false); cin.tie(NULL); cout << fixed << setprecision(15); } } io_setup; const int inf = (1 << 30) - 1; const ll INF = (1LL << 60) - 1; const int MOD = 1000000007; // const int MOD = 998244353; template struct Mod_Int { int x; Mod_Int() : x(0) {} Mod_Int(long long y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {} static int get_mod() { return mod; } Mod_Int &operator+=(const Mod_Int &p) { if ((x += p.x) >= mod) x -= mod; return *this; } Mod_Int &operator-=(const Mod_Int &p) { if ((x += mod - p.x) >= mod) x -= mod; return *this; } Mod_Int &operator*=(const Mod_Int &p) { x = (int)(1LL * x * p.x % mod); return *this; } Mod_Int &operator/=(const Mod_Int &p) { *this *= p.inverse(); return *this; } Mod_Int &operator++() { return *this += Mod_Int(1); } Mod_Int operator++(int) { Mod_Int tmp = *this; ++*this; return tmp; } Mod_Int &operator--() { return *this -= Mod_Int(1); } Mod_Int operator--(int) { Mod_Int tmp = *this; --*this; return tmp; } Mod_Int operator-() const { return Mod_Int(-x); } Mod_Int operator+(const Mod_Int &p) const { return Mod_Int(*this) += p; } Mod_Int operator-(const Mod_Int &p) const { return Mod_Int(*this) -= p; } Mod_Int operator*(const Mod_Int &p) const { return Mod_Int(*this) *= p; } Mod_Int operator/(const Mod_Int &p) const { return Mod_Int(*this) /= p; } bool operator==(const Mod_Int &p) const { return x == p.x; } bool operator!=(const Mod_Int &p) const { return x != p.x; } Mod_Int inverse() const { assert(*this != Mod_Int(0)); return pow(mod - 2); } Mod_Int pow(long long k) const { Mod_Int now = *this, ret = 1; for (; k > 0; k >>= 1, now *= now) { if (k & 1) ret *= now; } return ret; } friend ostream &operator<<(ostream &os, const Mod_Int &p) { return os << p.x; } friend istream &operator>>(istream &is, Mod_Int &p) { long long a; is >> a; p = Mod_Int(a); return is; } }; using mint = Mod_Int; template struct Matrix { vector> A; Matrix(int m, int n) : A(m, vector(n, 0)) {} int height() const { return A.size(); } int width() const { return A.front().size(); } inline const vector &operator[](int k) const { return A[k]; } inline vector &operator[](int k) { return A[k]; } static Matrix I(int l) { Matrix ret(l, l); for (int i = 0; i < l; i++) ret[i][i] = 1; return ret; } Matrix &operator*=(const Matrix &B) { int m = height(), n = width(), p = B.width(); assert(n == B.height()); Matrix ret(m, p); for (int i = 0; i < m; i++) { for (int k = 0; k < n; k++) { for (int j = 0; j < p; j++) ret[i][j] += A[i][k] * B[k][j]; } } swap(A, ret.A); return *this; } Matrix operator*(const Matrix &B) const { return Matrix(*this) *= B; } Matrix pow(long long k) const { int m = height(), n = width(); assert(m == n); Matrix now = *this, ret = I(n); for (; k > 0; k >>= 1, now *= now) { if (k & 1) ret *= now; } return ret; } bool eq(const T &a, const T &b) const { return a == b; // return abs(a-b) <= EPS; } pair row_reduction(vector &b) { //行基本変形を用いて簡約化を行い、(階数、行列式)の組を返す int m = height(), n = width(), check = 0, rank = 0; T det = 1; assert(b.size() == m); for (int j = 0; j < n; j++) { int pivot = check; for (int i = check; i < m; i++) { if (A[i][j] != 0) pivot = i; // if(abs(A[i][j]) > abs(A[pivot][j])) pivot = i; //Tが小数の場合はこちら } if (check != pivot) det *= T(-1); swap(A[check], A[pivot]), swap(b[check], b[pivot]); if (eq(A[check][j], T(0))) { det = T(0); continue; } rank++; det *= A[check][j]; T r = T(1) / A[check][j]; for (int k = j + 1; k < n; k++) A[check][k] *= r; b[check] *= r; A[check][j] = T(1); for (int i = 0; i < m; i++) { if (i == check) continue; if (!eq(A[i][j], 0)) { for (int k = j + 1; k < n; k++) A[i][k] -= A[i][j] * A[check][k]; b[i] -= A[i][j] * b[check]; } A[i][j] = T(0); } if (++check == m) break; } return make_pair(rank, det); } pair row_reduction() { vector b(height(), T(0)); return row_reduction(b); } Matrix inverse() { //行基本変形によって正方行列の逆行列を求める if (height() != width()) return Matrix(0, 0); int n = height(); Matrix ret = I(n); for (int j = 0; j < n; j++) { int pivot = j; for (int i = j; i < n; i++) { if (A[i][j] != 0) pivot = i; // if(abs(A[i][j]) > abs(A[pivot][j])) pivot = i; //Tが小数の場合はこちら } swap(A[j], A[pivot]), swap(ret[j], ret[pivot]); if (eq(A[j][j], T(0))) return Matrix(0, 0); T r = T(1) / A[j][j]; for (int k = j + 1; k < n; k++) A[j][k] *= r; for (int k = 0; k < n; k++) ret[j][k] *= r; A[j][j] = T(1); for (int i = 0; i < n; i++) { if (i == j) continue; if (!eq(A[i][j], T(0))) { for (int k = j + 1; k < n; k++) A[i][k] -= A[i][j] * A[j][k]; for (int k = 0; k < n; k++) ret[i][k] -= A[i][j] * ret[j][k]; } A[i][j] = T(0); } } return ret; } vector> Gausiann_elimination(vector b) { // Ax=bの解の1つと解空間の基底の組を返す int m = height(), n = width(); row_reduction(b); vector> ret; vector p(m, n); vector is_zero(n, true); for (int i = 0; i < m; i++) { for (int j = 0; j < n; j++) { if (!eq(A[i][j], T(0))) { p[i] = j; break; } } if (p[i] < n) is_zero[p[i]] = false; else if (!eq(b[i], T(0))) return {}; } vector x(n, T(0)); for (int i = 0; i < m; i++) { if (p[i] < n) x[p[i]] = b[i]; } ret.push_back(x); for (int j = 0; j < n; j++) { if (!is_zero[j]) continue; x[j] = T(1); for (int i = 0; i < m; i++) { if (p[i] < n) x[p[i]] = -A[i][j]; } ret.push_back(x), x[j] = T(0); } return ret; } }; int main() { int C, N, M; cin >> C >> N >> M; mint iv = mint(C).inverse(); using mat = Matrix; mat A(4, 4); rep(i, 4) { { if (i < 2) { A[i][i] += iv * iv; if (i == 0) { A[3][i] += iv * iv * (C - 1); } else { A[2][i] += iv * iv; A[3][i] += iv * iv * (C - 2); } } else { A[i % 2][i] += iv; } } { if (i % 2 == 0) { A[i + 1][i] += iv * (C - 1); } else { A[i][i] += iv * (C - 2); A[i - 1][i] += iv; } } } mat x(4, 4); x[0][0] = 1; A = A.pow(N / 2), A *= x; mint tmp = A[2 * (N & 1)][0]; if (N & 1) tmp += A[0][0] * iv; cout << mint(1) - mint(-tmp + 1).pow(M) << '\n'; }