#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math") #include using namespace std; #define rep(i, n) for (int i = 0; i < int(n); i++) #define per(i, n) for (int i = (n)-1; 0 <= i; i--) #define rep2(i, l, r) for (int i = (l); i < int(r); i++) #define per2(i, l, r) for (int i = (r)-1; int(l) <= i; i--) #define each(e, v) for (auto& e : v) #define MM << " " << #define pb push_back #define eb emplace_back #define all(x) begin(x), end(x) #define rall(x) rbegin(x), rend(x) #define sz(x) (int)x.size() 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' : ' '); if (v.empty()) cout << '\n'; } using ll = long long; using pii = 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 using minheap = std::priority_queue, std::greater>; template using maxheap = std::priority_queue; 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)); } // __int128_t gcd(__int128_t a, __int128_t b) { // if (a == 0) // return b; // if (b == 0) // return a; // __int128_t cnt = a % b; // while (cnt != 0) { // a = b; // b = cnt; // cnt = a % b; // } // return b; // } long long extGCD(long long a, long long b, long long& x, long long& y) { if (b == 0) { x = 1; y = 0; return a; } long long d = extGCD(b, a % b, y, x); y -= a / b * x; return d; } struct UnionFind { vector data; int num; UnionFind(int sz) { data.assign(sz, -1); num = sz; } bool unite(int x, int y) { x = find(x), y = find(y); if (x == y) return (false); if (data[x] > data[y]) swap(x, y); data[x] += data[y]; data[y] = x; num--; return (true); } int find(int k) { if (data[k] < 0) return (k); return (data[k] = find(data[k])); } int size(int k) { return (-data[find(k)]); } bool same(int x, int y) { return find(x) == find(y); } int operator[](int k) { return find(k); } }; 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; } }; ll mpow2(ll x, ll n, ll mod) { ll ans = 1; x %= mod; while (n != 0) { if (n & 1) ans = ans * x % mod; x = x * x % mod; n = n >> 1; } ans %= mod; return ans; } ll modinv2(ll a, ll mod) { ll b = mod, u = 1, v = 0; while (b) { ll t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v); } u %= mod; if (u < 0) u += mod; return u; } ll divide_int(ll a, ll b) { if (b < 0) a = -a, b = -b; return (a >= 0 ? a / b : (a - b + 1) / b); } // const int MOD = 1000000007; const int MOD = 998244353; using mint = Mod_Int; mint mpow(mint x, ll n) { bool rev = n < 0; n = abs(n); mint ans = 1; while (n != 0) { if (n & 1) ans *= x; x *= x; n = n >> 1; } return (rev ? ans.inverse() : ans); } // ----- library ------- // ----- library ------- int main() { ios::sync_with_stdio(false); std::cin.tie(nullptr); cout << fixed << setprecision(15); int n; string s; cin >> n >> s; s += '#'; vector f(n + 1, 0), g0(n + 3, 0), g1(n + 3, 0); double h0 = 0, h1 = 0; per(i, n + 1) { if (s[i] == '#') g1[i] = 1; else if (i + 1 != n) { f[i] = (h0 + 3) / (2 - h1); g0[i] = (h0 + f[i]) / 3 + 1; g1[i] = h1 / 3; } h0 += g0[i] - g0[i + 2], h1 += g1[i] - g1[i + 2]; } cout << f[0] << endl; }