#include namespace { #pragma GCC diagnostic ignored "-Wunused-function" #include #pragma GCC diagnostic warning "-Wunused-function" using namespace std; using namespace atcoder; #define rep(i,n) for(int i = 0; i < (int)(n); i++) #define rrep(i,n) for(int i = (int)(n) - 1; i >= 0; i--) #define all(x) begin(x), end(x) #define rall(x) rbegin(x), rend(x) template bool chmax(T& a, const T& b) { if (a < b) { a = b; return true; } else return false; } template bool chmin(T& a, const T& b) { if (b < a) { a = b; return true; } else return false; } using ll = long long; using P = pair; using VI = vector; using VVI = vector; using VL = vector; using VVL = vector; using mint = modint; } int main() { ios::sync_with_stdio(false); cin.tie(0); int n, a, b, c; cin >> n >> a >> b >> c; int fact_int[10]; fact_int[0] = 1; for (int i = 1; i < 10; i++) fact_int[i] = fact_int[i-1] * i; mint::set_mod(n); vector fact(n); fact[0] = 1; for (int i = 1; i < n; i++) fact[i] = fact[i-1] * i; int ub = 2 * a + 4 * c; VI dp(n, ub); priority_queue, greater

> q; auto add = [&](int u, int cost) { if (chmin(dp[u], cost)) q.emplace(cost, u); }; add(1, 0); while (q.size()) { auto [cost, u] = q.top(); q.pop(); add((u + 1) % n, cost + a); { int bk = b; mint uk = u; while (!__builtin_mul_overflow(bk, b, &bk) && bk < ub) { uk *= u; add(uk.val(), cost + bk); } } add(fact[u].val(), cost + c); } int ans = dp[0]; rep(i, n) if (i >= 10 || fact_int[i] >= n) chmin(ans, dp[i] + 2 * c); for (int x = 2; x < n; x++) { int bk = b; int xk = x; while (true) { if (__builtin_mul_overflow(xk, x, &xk)) xk = n; if (__builtin_mul_overflow(bk, b, &bk) || bk >= ub) break; if (xk >= n) { chmin(ans, dp[x] + bk + c); break; } } } cout << ans << '\n'; }