#pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; //#define int long long typedef long long ll; typedef unsigned long long ul; typedef unsigned int ui; //ll mod = 1; constexpr ll mod = 998244353; //constexpr ll mod = 1000000009; const int mod17 = 1000000007; const ll INF = (ll)mod17 * mod17; typedef pairP; #define rep(i,n) for(int i=0;i=0;i--) #define Rep(i,sta,n) for(int i=sta;i=1;i--) #define Rep1(i,sta,n) for(int i=sta;i<=n;i++) #define all(v) (v).begin(),(v).end() typedef pair LP; using ld = double; typedef pair LDP; const ld eps = 1e-10; const ld pi = acosl(-1.0); template void chmin(T& a, T b) { a = min(a, b); } template void chmax(T& a, T b) { a = max(a, b); } template vector vmerge(vector& a, vector& b) { vector res; int ida = 0, idb = 0; while (ida < a.size() || idb < b.size()) { if (idb == b.size()) { res.push_back(a[ida]); ida++; } else if (ida == a.size()) { res.push_back(b[idb]); idb++; } else { if (a[ida] < b[idb]) { res.push_back(a[ida]); ida++; } else { res.push_back(b[idb]); idb++; } } } return res; } template void cinarray(vector& v) { rep(i, v.size())cin >> v[i]; } template void coutarray(vector& v) { rep(i, v.size()) { if (i > 0)cout << " "; cout << v[i]; } cout << "\n"; } ll mod_pow(ll x, ll n, ll m = mod) { if (n < 0) { ll res = mod_pow(x, -n, m); return mod_pow(res, m - 2, m); } if (abs(x) >= m)x %= m; if (x < 0)x += m; //if (x == 0)return 0; ll res = 1; while (n) { if (n & 1)res = res * x % m; x = x * x % m; n >>= 1; } return res; } //mod should be <2^31 struct modint { int n; modint() :n(0) { ; } modint(ll m) { if (m < 0 || mod <= m) { m %= mod; if (m < 0)m += mod; } n = m; } operator int() { return n; } }; bool operator==(modint a, modint b) { return a.n == b.n; } bool operator<(modint a, modint b) { return a.n < b.n; } modint operator+=(modint& a, modint b) { a.n += b.n; if (a.n >= mod)a.n -= (int)mod; return a; } modint operator-=(modint& a, modint b) { a.n -= b.n; if (a.n < 0)a.n += (int)mod; return a; } modint operator*=(modint& a, modint b) { a.n = ((ll)a.n * b.n) % mod; return a; } modint operator+(modint a, modint b) { return a += b; } modint operator-(modint a, modint b) { return a -= b; } modint operator*(modint a, modint b) { return a *= b; } modint operator^(modint a, ll n) { if (n == 0)return modint(1); modint res = (a * a) ^ (n / 2); if (n % 2)res = res * a; return res; } ll inv(ll a, ll p) { return (a == 1 ? 1 : (1 - p * inv(p % a, a)) / a + p); } modint operator/(modint a, modint b) { return a * modint(inv(b, mod)); } modint operator/=(modint& a, modint b) { a = a / b; return a; } const int max_n = 1 << 20; modint fact[max_n], factinv[max_n]; void init_f() { fact[0] = modint(1); for (int i = 0; i < max_n - 1; i++) { fact[i + 1] = fact[i] * modint(i + 1); } factinv[max_n - 1] = modint(1) / fact[max_n - 1]; for (int i = max_n - 2; i >= 0; i--) { factinv[i] = factinv[i + 1] * modint(i + 1); } } modint comb(int a, int b) { if (a < 0 || b < 0 || a < b)return 0; return fact[a] * factinv[b] * factinv[a - b]; } modint combP(int a, int b) { if (a < 0 || b < 0 || a < b)return 0; return fact[a] * factinv[a - b]; } ll gcd(ll a, ll b) { a = abs(a); b = abs(b); if (a < b)swap(a, b); while (b) { ll r = a % b; a = b; b = r; } return a; } template void addv(vector& v, int loc, T val) { if (loc >= v.size())v.resize(loc + 1, 0); v[loc] += val; } /*const int mn = 2000005; bool isp[mn]; vector ps; void init() { fill(isp + 2, isp + mn, true); for (int i = 2; i < mn; i++) { if (!isp[i])continue; ps.push_back(i); for (int j = 2 * i; j < mn; j += i) { isp[j] = false; } } }*/ //[,val) template auto prev_itr(set& st, T val) { auto res = st.lower_bound(val); if (res == st.begin())return st.end(); res--; return res; } //[val,) template auto next_itr(set& st, T val) { auto res = st.lower_bound(val); return res; } using mP = pair; mP operator+(mP a, mP b) { return { a.first + b.first,a.second + b.second }; } mP operator+=(mP& a, mP b) { a = a + b; return a; } mP operator-(mP a, mP b) { return { a.first - b.first,a.second - b.second }; } mP operator-=(mP& a, mP b) { a = a - b; return a; } LP operator+(LP a, LP b) { return { a.first + b.first,a.second + b.second }; } LP operator+=(LP& a, LP b) { a = a + b; return a; } LP operator-(LP a, LP b) { return { a.first - b.first,a.second - b.second }; } LP operator-=(LP& a, LP b) { a = a - b; return a; } mt19937 mt(time(0)); const string drul = "DRUL"; string senw = "SENW"; //DRUL,or SENW //int dx[4] = { 1,0,-1,0 }; //int dy[4] = { 0,1,0,-1 }; //------------------------------------ int get_premitive_root() { int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { set fac; int v = mod - 1; for (ll i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < mod; g++) { bool ok = true; for (auto i : fac) if (mod_pow(g, (mod - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } const int proot = get_premitive_root(); int bsf(int x) { int res = 0; while (!(x & 1)) { res++; x >>= 1; } return res; } int ceil_pow2(int n) { int x = 0; while ((1 << x) < n) x++; return x; } using poly = vector; void butterfly(poly& a) { int n = int(a.size()); int g = proot; int h = ceil_pow2(n); static bool first = true; static modint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i] if (first) { first = false; modint es[30], ies[30]; // es[i]^(2^(2+i)) == 1 int cnt2 = bsf(mod - 1); modint e = mod_pow(g, (mod - 1) >> cnt2); modint ie = (modint)1 / e; for (int i = cnt2; i >= 2; i--) { // e^(2^i) == 1 es[i - 2] = e; ies[i - 2] = ie; e *= e; ie *= ie; } modint now = 1; for (int i = 0; i < cnt2 - 2; i++) { sum_e[i] = es[i] * now; now *= ies[i]; } } for (int ph = 1; ph <= h; ph++) { int w = 1 << (ph - 1), p = 1 << (h - ph); modint now = 1; for (int s = 0; s < w; s++) { int offset = s << (h - ph + 1); for (int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p] * now; a[i + offset] = l + r; a[i + offset + p] = l - r; } now *= sum_e[bsf(~(unsigned int)(s))]; } } } void butterfly_inv(poly& a) { int n = int(a.size()); int g = proot; int h = ceil_pow2(n); static bool first = true; static modint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i] if (first) { first = false; modint es[30], ies[30]; // es[i]^(2^(2+i)) == 1 int cnt2 = bsf(mod - 1); modint e = mod_pow(g, (mod - 1) >> cnt2); modint ie = (modint)1 / e; for (int i = cnt2; i >= 2; i--) { // e^(2^i) == 1 es[i - 2] = e; ies[i - 2] = ie; e *= e; ie *= ie; } modint now = 1; for (int i = 0; i < cnt2 - 2; i++) { sum_ie[i] = ies[i] * now; now *= es[i]; } } for (int ph = h; ph >= 1; ph--) { int w = 1 << (ph - 1), p = 1 << (h - ph); modint inow = 1; for (int s = 0; s < w; s++) { int offset = s << (h - ph + 1); for (int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p]; a[i + offset] = l + r; a[i + offset + p] = (unsigned long long)(mod + (ll)l - (ll)r) * (ll)inow; } inow *= sum_ie[bsf(~(unsigned int)(s))]; } } } poly multiply(poly g, poly h) { int n = g.size(); int m = h.size(); if (n == 0 || m == 0)return {}; if (min(g.size(), h.size()) < 60) { poly res(g.size() + h.size() - 1); rep(i, g.size())rep(j, h.size()) { res[i + j] += g[i] * h[j]; } return res; } int z = 1 << ceil_pow2(n + m - 1); g.resize(z); butterfly(g); h.resize(z); butterfly(h); rep(i, z) { g[i] *= h[i]; } butterfly_inv(g); g.resize(n + m - 1); modint iz = (modint)1 / (modint)z; rep(i, n + m - 1) { g[i] *= iz; } return g; } struct FormalPowerSeries :vector { using vector::vector; using fps = FormalPowerSeries; void shrink() { while (this->size() && this->back() == (modint)0)this->pop_back(); } fps operator+(const fps& r)const { return fps(*this) += r; } fps operator+(const modint& v)const { return fps(*this) += v; } fps operator-(const fps& r)const { return fps(*this) -= r; } fps operator-(const modint& v)const { return fps(*this) -= v; } fps operator*(const fps& r)const { return fps(*this) *= r; } fps operator*(const modint& v)const { return fps(*this) *= v; } fps& operator+=(const fps& r) { if (r.size() > this->size())this->resize(r.size()); rep(i, r.size())(*this)[i] += r[i]; shrink(); return *this; } fps& operator+=(const modint& v) { if (this->empty())this->resize(1); (*this)[0] += v; shrink(); return *this; } fps& operator-=(const fps& r) { if (r.size() > this->size())this->resize(r.size()); rep(i, r.size())(*this)[i] -= r[i]; shrink(); return *this; } fps& operator-=(const modint& v) { if (this->empty())this->resize(1); (*this)[0] -= v; shrink(); return *this; } fps& operator*=(const fps& r) { if (this->empty() || r.empty())this->clear(); else { poly ret = multiply(*this, r); *this = fps(all(ret)); } shrink(); return *this; } fps& operator*=(const modint& v) { for (auto& x : (*this))x *= v; shrink(); return *this; } fps operator-()const { fps ret = *this; for (auto& v : ret)v = -v; return ret; } modint sub(modint x) { modint t = 1; modint res = 0; rep(i, (*this).size()) { res += t * (*this)[i]; t *= x; } return res; } fps pre(int sz)const { fps ret(this->begin(), this->begin() + min((int)this->size(), sz)); ret.shrink(); return ret; } fps integral() const { const int n = (int)this->size(); fps ret(n + 1); ret[0] = 0; for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / (modint)(i + 1); return ret; } fps inv(int deg = -1)const { const int n = this->size(); if (deg == -1)deg = n; fps ret({ (modint)1 / (*this)[0] }); for (int i = 1; i < deg; i <<= 1) { ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1); } ret = ret.pre(deg); ret.shrink(); return ret; } fps diff() const { const int n = (int)this->size(); fps ret(max(0, n - 1)); for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * (modint)i; return ret; } // F(0) must be 1 fps log(int deg = -1) const { assert((*this)[0] == 1); const int n = (int)this->size(); if (deg == -1) deg = n; return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } // F(0) must be 0 fps exp(int deg = -1)const { assert((*this)[0] == 0); const int n = (int)this->size(); if (deg == -1)deg = n; fps ret = { 1 }; for (int i = 1; i < deg; i <<= 1) { ret = (ret * (pre(i << 1) + 1 - ret.log(i << 1))).pre(i << 1); } //cout << "!!!! " << ret.size() << "\n"; return ret.pre(deg); } fps div(fps g) { assert(g.size() && g.back() != (modint)0); fps f = *this; if (f.size() < g.size())return {}; int dif = f.size() - g.size(); reverse(all(f)); reverse(all(g)); g = g.inv(dif + 1); fps fg = f * g; fps ret(dif + 1); rep(i, fg.size()) { int id = i - dif; if (-dif <= id && id <= 0) { ret[-id] = fg[i]; } } return ret; } fps divr(fps g) { fps ret = (*this) - g * (*this).div(g); ret.shrink(); return ret; } }; using fps = FormalPowerSeries; //f(r^0),f(r^1),...,f(r^n) vector Multipoint_Evaluation(fps c, modint r, int n) { vector res(n + 1, 0); if (c.empty()) { return res; } if (r == (modint)0) { rep(i, n + 1)res[i] = c[0]; return res; } int sz = c.size() + n; vector rr(sz); rr[0] = 1; rep(i, sz - 1)rr[i + 1] = rr[i] * r; vector irr(sz); modint ir = (modint)1 / r; irr[0] = 1; rep(i, sz - 1)irr[i + 1] = irr[i] * ir; vector coef(sz); coef[0] = 1; rep(i, sz - 1) { coef[i + 1] = coef[i] * rr[i]; } vector icoef(sz); icoef[0] = 1; rep(i, sz - 1) { icoef[i + 1] = icoef[i] * irr[i]; } fps f(c.size()); rep(i, c.size()) { f[i] = (modint)c[i] * icoef[i]; } fps g(sz); rep(i, sz) { g[i] = coef[i]; } reverse(all(f)); f *= g; rep(i, n + 1) { modint val = icoef[i]; int loc = i + c.size() - 1; if (loc < f.size())val *= f[loc]; else val = 0; res[i] = val; } return res; } vector Multipoint_Evaluation(fps c, vector p) { int n = p.size(); vector ret(n); int sz = 1; while (sz < n)sz *= 2; vector f(2 * sz - 1); function dfs = [&](int k, int l, int r) { if (l + 1 == r) { f[k] = { -p[l],1 }; } else { dfs(2 * k + 1, l, (l + r) / 2); dfs(2 * k + 2, (l + r) / 2, r); f[k] = f[2 * k + 1] * f[2 * k + 2]; } }; dfs(0, 0, n); vector g(2 * sz - 1); function invdfs = [&](int k, int l, int r) { if (k == 0) { g[k] = c.divr(f[k]); } else { g[k] = g[(k - 1) / 2].divr(f[k]); } if (r - l <= 100) { Rep(i, l, r) { ret[i] = g[k].sub(p[i]); } } else { invdfs(2 * k + 1, l, (l + r) / 2); invdfs(2 * k + 2, (l + r) / 2, r); } }; invdfs(0, 0, n); return ret; } //reference: https://37zigen.com/berlekamp-massey/ struct berlekamp_massey { fps a, b; berlekamp_massey(int n, fps f) { f.resize(2 * n); f.shrink(); //deg(a) c2.size()) { swap(c1, c2); swap(a1, a2); swap(b1, b2); } if (c1.size() <= n)break; int dif = c2.size() - c1.size(); modint coef = c2.back() / c1.back(); fps d1, d2, d3; d1.resize(dif); d2.resize(dif); d3.resize(dif); rep(i, a1.size())d1.push_back(a1[i] * coef); rep(i, b1.size())d2.push_back(b1[i] * coef); rep(i, c1.size())d3.push_back(c1[i] * coef); a2 -= d1; b2 -= d2; c2 -= d3; } swap(a, c1); swap(b, a1); } //g=x^{-n}*a mod b ll calc(ll n) { assert(b[0] != (modint)0); modint coef = (modint)1 / b[0]; rep(i, a.size())a[i] *= coef; rep(i, b.size())b[i] *= coef; fps rx = b; coef = (modint)-1 / b[0]; rx.erase(rx.begin()); rep(i, rx.size())rx[i] *= coef; //rx^n fps z = a; while (n) { if (n & 1) { z *= rx; z = z.divr(b); } n >>= 1; if (n == 0)break; rx *= rx; rx = rx.divr(b); } if (z.empty())return 0; return z[0] / b[0]; } }; fps allprod(vector f) { while (f.size() > 1) { vector nf; for (int i = 0; i + 1 < f.size(); i += 2) { nf.push_back(f[i] * f[i + 1]); } if (f.size() % 2)nf.push_back(f.back()); swap(f, nf); } return f[0]; } //https://maspypy.com/%e5%a4%9a%e9%a0%85%e5%bc%8f%e3%83%bb%e5%bd%a2%e5%bc%8f%e7%9a%84%e3%81%b9%e3%81%8d%e7%b4%9a%e6%95%b0-%e9%ab%98%e9%80%9f%e3%81%ab%e8%a8%88%e7%ae%97%e3%81%a7%e3%81%8d%e3%82%8b%e3%82%82%e3%81%ae#toc40 //sum_{i}(a_i*exp(b_ix))をm次まで計算 //verified with https://atcoder.jp/contests/arc154/tasks/arc154_f fps calc_expsum(vector ps, int m) { using pfps = pair; vector vp; rep(i, ps.size()) { modint a = ps[i].first; modint b = ps[i].second; fps pa = { a }; fps pb = { 1,-b }; vp.push_back({ pa,pb }); } while (vp.size() > 1) { vector nvp; for (int i = 0; i + 1 < vp.size(); i += 2) { pfps& a = vp[i]; pfps& b = vp[i + 1]; fps pa = a.first * b.second + a.second * b.first; if (pa.size() > m + 1)pa.resize(m + 1); fps pb = a.second * b.second; if (pb.size() > m + 1)pb.resize(m + 1); nvp.push_back({ pa,pb }); } if (vp.size() & 1)nvp.push_back(vp.back()); swap(vp, nvp); } fps las = vp[0].first * vp[0].second.inv(m + 1); fps res(m + 1); rep(i, m + 1) { if (i < las.size())res[i] = las[i]; res[i] *= factinv[i]; } return res; } fps GetBernoulli(int n,bool rev) { fps f(n + 2); rep(i, n + 2) { f[i] = factinv[i]; } f.erase(f.begin()); f = f.inv(n + 1); f.resize(n + 1); rep(i, n + 1)f[i] *= fact[i]; if (rev) { for (int i = 1; i <= n; i += 2)f[i] *= -1; } return f; } fps GetPrefixSumPolynomial(fps p) { int n = p.size(); fps ber = GetBernoulli(n,true); fps a(n); rep(i, n)a[i] = p[i] * fact[i]; a[0] = 0; fps b(n); rep(i, n)b[n - 1 - i] = ber[i] * factinv[i]; a *= b; fps res(n + 1); rep(i, n) { int loc = i + n - 1; modint val = 0; if (loc < a.size())val = a[loc]; val *= factinv[i + 1]; res[i + 1] = val; } res[0] += p[0]; res[1] += p[0]; return res; } //f(x)->f(ax+b) fps shift(fps f,modint a,modint b) { int n = f.size(); vector rb(n); rb[0] = 1; rep(i, n - 1)rb[i + 1] = rb[i] * b; fps x(n); rep(i, n)x[i] = f[i] * fact[i]; fps y(n); rep(i, n)y[n - 1 - i] = factinv[i] * rb[i]; x *= y; fps res(n); rep(i, n) { int loc = i + n - 1; modint val = 0; if (loc < x.size())val = x[loc]; val *= factinv[i]; res[i] = val; } if (a != 1) { modint pro = 1; rep(i, res.size()) { res[i] *= pro; pro *= a; } } return res; } void expr() { uniform_int_distribution<> udn(2,20); uniform_int_distribution<> udx(1, 1<<20); rep(t, 1000) { int n = udn(mt); fps f(n); rep(i, n)f[i] = udx(mt); fps g = GetPrefixSumPolynomial(f); auto calc = [&](fps& f, modint x) { modint pro = 1; modint res = 0; rep(i, f.size()) { res += pro * f[i]; pro *= x; } return res; }; modint sum = 0; rep(i, 100) { sum += calc(f, i); if (sum != calc(g, i)) { cout << t << " " << n << "\n"; coutarray(f); coutarray(g); exit(0); } assert(sum == calc(g, i)); } } cout << "fin\n"; } void solve() { int n; cin >> n; n--; vector a(n); rep(i, n)cin >> a[i]; string s; cin >> s; fps cur = { 1 }; rep(i, n) { int r = 0; string ns; rep(j, s.size()) { r = 10 * r + (s[j] - '0'); ns.push_back('0' + r / a[i]); r %= a[i]; } swap(s, ns); cur = shift(cur, a[i], r); cur = GetPrefixSumPolynomial(cur); } modint x = 0; rep(i, s.size())x = (modint)10 * x + (modint)(s[i] - '0'); modint ans = cur.sub(x); cout << ans << "\n"; } signed main() { ios::sync_with_stdio(false); cin.tie(0); //cout << fixed<> t; rep(i, t) solve(); return 0; }