#include using namespace std; using ll=long long; using ld=long double; using ull=unsigned long long; using uint=unsigned int; using pcc=pair; using pii=pair; using pll=pair; using pdd=pair; using tuplis=pair; using tuplis2=pair; template using pq=priority_queue,greater>; const ll LINF=0x1fffffffffffffff; const ll MINF=0x7fffffffffff; const int INF=0x3fffffff; const ll MOD=1000000007; const ll MODD=998244353; const ld DINF=numeric_limits::infinity(); const ld EPS=1e-9; const ld PI=3.141592653589793238462643383279; const vectorfour{0,1,0,-1,0}; #define _overload4(_1,_2,_3,_4,name,...) name #define _overload3(_1,_2,_3,name,...) name #define _rep1(n) for(ll i=0;i=0;i--) #define _rrep2(i,n) for(ll i=(n)-1;i>=0;i--) #define _rrep3(i,a,b) for(ll i=(b)-1;i>=(a);i--) #define _rrep4(i,a,b,c) for(ll i=a+(b-a-1)/c*c;i>=a;i-=c) #define rrep(...) _overload4(__VA_ARGS__,_rrep4,_rrep3,_rrep2,_rrep1)(__VA_ARGS__) #define each(i,a) for(auto &i:a) #define sum(...) accumulate(range(__VA_ARGS__),0LL) #define dsum(...) accumulate(range(__VA_ARGS__),double(0)) #define _range(i) (i).begin(),(i).end() #define _range2(i,k) (i).begin(),(i).begin()+k #define _range3(i,a,b) (i).begin()+a,(i).begin()+b #define range(...) _overload3(__VA_ARGS__,_range3,_range2,_range)(__VA_ARGS__) #define _rrange(i) (i).rbegin(),(i).rend() #define _rrange2(i,k) (i).rbegin(),(i).rbegin()+k #define _rrange3(i,a,b) (i).rbegin()+a,(i).rbegin()+b #define rrange(...) _overload3(__VA_ARGS__,_rrange3,_rrange2,_rrange)(__VA_ARGS__) #define elif else if #define unless(a) if(!(a)) #define mp make_pair #define mt make_tuple #define INT(...) int __VA_ARGS__;in(__VA_ARGS__) #define LL(...) ll __VA_ARGS__;in(__VA_ARGS__) #define ULL(...) ull __VA_ARGS__;in(__VA_ARGS__) #define STR(...) string __VA_ARGS__;in(__VA_ARGS__) #define CHR(...) char __VA_ARGS__;in(__VA_ARGS__) #define DBL(...) double __VA_ARGS__;in(__VA_ARGS__) #define LD(...) ld __VA_ARGS__;in(__VA_ARGS__) #define vec(type,name,...) vector name(__VA_ARGS__) #define VEC(type,name,size) vector name(size);in(name) #define vv(type,name,h,...) vector>name(h,vector(__VA_ARGS__)) #define VV(type,name,h,...) vector>name(h,vector(__VA_ARGS__));in(name) #define vvv(type,name,h,w,...) vector>>name(h,vector>(w,vector(__VA_ARGS__))) inline ll gcd(ll a, ll b){ while(b){ ll c = b; b = a % b; a = c; } return a; } inline ll lcm(ll a, ll b){ if(!a || !b) return 0; return a * b / gcd(a, b); } template inline T min(const vector &v){ return *min_element(range(v)); } inline char min(const string &v){ return *min_element(range(v)); } template inline T max(const vector &v){ return *max_element(range(v)); } inline char max(const string &v){ return *max_element(range(v)); } inline ll intpow(ll a, ll b){ ll ans = 1; while(b){ if(b & 1) ans *= a; a *= a; b /= 2; } return ans; } inline ll modpow(ll a, ll b, ll p){ ll ans = 1; while(b){ if(b & 1) (ans *= a) %= p; (a *= a) %= p; b /= 2; } return ans; } template inline bool chmin(T &a, const T &b){ if(a > b){ a = b; return 1; } return 0; } template inline bool chmax(T &a, const T &b){ if(a < b){ a = b; return 1; } return 0; } map factor(ull x){ map ans; for(ll i = 2; i * i <= x; i++) while(x % i == 0){ ans[i]++; x /= i; } if(x != 1) ans[x]++; return ans; } set divisor(ull x){ set ans; for(ll i = 1; i * i <= x; i++) if(x % i == 0){ ans.insert(i); ans.insert(x / i); } return ans; } template unordered_map press(vector &a){ auto b = a; sort(range(b)); b.erase(unique(range(b)), b.end()); unordered_map ans; rep(b.size()) ans[b[i]] = i; each(i, a) i = ans[i]; return ans; } template map press_map(vector &a){ auto b = a; sort(range(b)); b.erase(unique(range(b)), b.end()); map ans; rep(b.size()) ans[b[i]] = i; each(i, a) i = ans[i]; return ans; } inline int scan(){ return getchar(); } inline void scan(int &a){ scanf("%d", &a); } inline void scan(unsigned &a){ scanf("%u", &a); } inline void scan(long &a){ scanf("%ld", &a); } inline void scan(long long &a){ scanf("%lld", &a); } inline void scan(unsigned long long &a){ scanf("%llu", &a); } inline void scan(char &a){ cin >> a; } inline void scan(float &a){ scanf("%f", &a); } inline void scan(double &a){ scanf("%lf", &a); } inline void scan(long double &a){ scanf("%Lf", &a); } inline void scan(vector &vec){ for(unsigned i = 0; i < vec.size(); i++) { int a; scan(a); vec[i] = a; } } inline void scan(char a[]){ scanf("%s", a); } inline void scan(string &a){ cin >> a; } template inline void scan(vector &vec); template inline void scan(array &vec); template inline void scan(pair &p); template inline void scan(T (&vec)[size]); template inline void scan(vector &vec){ for(auto &i : vec) scan(i); } template inline void scan(deque &vec){ for(auto &i : vec) scan(i); } template inline void scan(array &vec){ for(auto &i : vec) scan(i); } template inline void scan(pair &p){ scan(p.first); scan(p.second); } template inline void scan(T (&vec)[size]){ for(auto &i : vec) scan(i); } template inline void scan(T &a){ cin >> a; } inline void in(){} template inline void in(Head &head, Tail&... tail){ scan(head); in(tail...); } inline void print(){ putchar(' '); } inline void print(const bool &a){ printf("%d", a); } inline void print(const int &a){ printf("%d", a); } inline void print(const unsigned &a){ printf("%u", a); } inline void print(const long &a){ printf("%ld", a); } inline void print(const long long &a){ printf("%lld", a); } inline void print(const unsigned long long &a){ printf("%llu", a); } inline void print(const char &a){ printf("%c", a); } inline void print(const char a[]){ printf("%s", a); } inline void print(const float &a){ printf("%.15f", a); } inline void print(const double &a){ printf("%.15f", a); } inline void print(const long double &a){ printf("%.15Lf", a); } inline void print(const string &a){ for(auto&& i : a) print(i); } template inline void print(const vector &vec); template inline void print(const array &vec); template inline void print(const pair &p); template inline void print(const T (&vec)[size]); template inline void print(const vector &vec){ if(vec.empty()) return; print(vec[0]); for(auto i = vec.begin(); ++i != vec.end(); ){ putchar(' '); print(*i); } } template inline void print(const deque &vec){ if(vec.empty()) return; print(vec[0]); for(auto i = vec.begin(); ++i != vec.end(); ){ putchar(' '); print(*i); } } template inline void print(const array &vec){ print(vec[0]); for(auto i = vec.begin(); ++i != vec.end(); ){ putchar(' '); print(*i); } } template inline void print(const pair &p){ print(p.first); putchar(' '); print(p.second); } template inline void print(const T (&vec)[size]){ print(vec[0]); for(auto i = vec; ++i != end(vec); ){ putchar(' '); print(*i); } } template inline void print(const T &a){ cout << a; } inline int out(){ putchar('\n'); return 0; } template inline int out(const T &t){ print(t); putchar('\n'); return 0; } template inline int out(const Head &head, const Tail&... tail){ print(head); putchar(' '); out(tail...); return 0; } #ifdef DEBUG inline void err(){ putchar('\n'); } template inline void err(const T &t){ print(t); putchar('\n'); } template inline void err(const Head &head, const Tail&... tail){ print(head); putchar(' '); out(tail...); } #else template inline void err(const T&...){} #endif inline int first(bool i = true){ return out(i?"first":"second"); } inline int yes(bool i = true){ return out(i?"yes":"no"); } inline int Yes(bool i = true){ return out(i?"Yes":"No"); } inline int No(){ return out("No"); } inline int YES(bool i = true){ return out(i?"YES":"NO"); } inline int NO(){ return out("NO"); } inline int Yay(bool i = true){ return out(i?"Yay!":":("); } inline int Possible(bool i = true){ return out(i?"Possible":"Impossible"); } inline int POSSIBLE(bool i = true){ return out(i?"POSSIBLE":"IMPOSSIBLE"); } inline void Case(ll i){ printf("Case #%lld: ", i); } constexpr ll mod = MODD; inline constexpr ll extgcd(ll a, ll b, ll &x, ll &y){ ll g = a; x = 1; y = 0; if(b){ g = extgcd(b, a % b, y, x); y -= a / b * x; } return g; } inline constexpr ll invmod(ll a, ll m = mod){ ll x = 0, y = 0; extgcd(a, m, x, y); return (x + m) % m; } struct Modint{ ll num; constexpr Modint() : num() { num = 0; } constexpr Modint(ll x) : num() { num = x % mod; if(num < 0) num += mod; } inline constexpr Modint& operator= (int x){ num = x % mod; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator= (ll x){ num = x % mod; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator= (Modint x){ num = x.num; return *this; } inline constexpr Modint operator+ (int x) const { return Modint(num + x); } inline constexpr Modint operator+ (ll x) const { return Modint(num + x); } inline constexpr Modint operator+ (Modint x) const { ll a = num + x.num; if(a >= mod) a -= mod; return Modint{a}; } inline constexpr Modint& operator+=(int x){ num += x; num %= mod; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator+=(ll x){ num += x; num %= mod; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator+=(Modint x){ num += x.num; if(num >= mod) num -= mod; return *this; } inline constexpr Modint& operator++(){ if(num == mod - 1) num = 0; else num++; return *this; } inline constexpr Modint operator++(int){ Modint ans = *this; if(num == mod - 1) num = 0; else num++; return ans; } inline constexpr Modint operator- () const { if(num == 0) return 0; return Modint(mod - num); } inline constexpr Modint operator- (int x) const { return Modint(num - x); } inline constexpr Modint operator- (ll x) const { return Modint(num - x); } inline constexpr Modint operator- (Modint x) const { ll a = num - x.num; if(a < 0) a += mod; return Modint{a}; } inline constexpr Modint& operator-=(int x){ num -= x; num %= mod; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator-=(ll x){ num -= x; num %= mod; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator-=(Modint x){ num -= x.num; if(num < 0) num += mod; return *this; } inline constexpr Modint& operator--(){ if(num == 0) num = mod - 1; else num--; return *this; } inline constexpr Modint operator--(int){ Modint ans = *this; if(num == 0) num = mod - 1; else num--; return ans; } inline constexpr Modint operator* (int x) const { return Modint(num * (x % mod)); } inline constexpr Modint operator* (ll x) const { return Modint(num * (x % mod)); } inline constexpr Modint operator* (Modint x) const { return Modint{num * x.num % mod}; } inline constexpr Modint& operator*=(int x){ num *= Modint(x); num %= mod; return *this; } inline constexpr Modint& operator*=(ll x){ num *= Modint(x); num %= mod; return *this; } inline constexpr Modint& operator*=(Modint x){ num *= x.num; num %= mod; return *this; } inline constexpr Modint operator/ (int x) const { return Modint(num * invmod(Modint(x), mod)); } inline constexpr Modint operator/ (ll x) const { return Modint(num * invmod(Modint(x), mod)); } inline constexpr Modint operator/ (Modint x) const { return Modint{num * invmod(x.num, mod) % mod}; } inline constexpr Modint& operator/=(int x){ num *= invmod(Modint(x), mod); num %= mod; return *this; } inline constexpr Modint& operator/=(ll x){ num *= invmod(Modint(x), mod); num %= mod; return *this; } inline constexpr Modint& operator/=(Modint x){ num *= invmod(x.num, mod); num %= mod; return *this; } inline constexpr Modint pow(ll x) const { ll i = 1; Modint ans = 1, cnt = *this; while(i <= x){ if(x & i){ ans *= cnt; x ^= i; } cnt *= cnt; i *= 2; } return ans; } inline constexpr operator ll() const { return num; } }; inline constexpr Modint operator""_M(ull x) { return Modint(x); } std::vector fac(1, 1), inv(1, 1); inline void reserve(ll a){ if(fac.size() >= a) return; if(a < fac.size() * 2) a = fac.size() * 2; if(a >= mod) a = mod; while(fac.size() < a) fac.push_back(fac.back() * ll(fac.size())); inv.resize(fac.size()); inv.back() = Modint(1) / fac.back(); for(ll i = inv.size() - 1; !inv[i - 1]; i--) inv[i - 1] = inv[i] * i; } inline Modint fact(ll n){ if(n < 0) return 0; reserve(n + 1); return fac[n]; } inline Modint perm(ll n, ll r){ if(r < 0 || n < r) return 0; reserve(n + 1); return fac[n] * inv[n - r]; } inline Modint comb(ll n, ll r){ if(r < 0 || n < r) return 0; reserve(n + 1); return fac[n] * inv[r] * inv[n - r]; } inline Modint Mcomb(ll n, ll r){ return comb(n + r - 1, n - 1); } // r個をn部屋に分ける istream& operator>>(istream& is, Modint& x) { is >> x.num; x = Modint(x.num); return is; } struct FFT{ size_t size = 1, size_log2 = 0; vector butterfly; vector zeta; FFT(size_t n) { // 掛け算後のsize while(size < n) { size *= 2; size_log2++; } butterfly.resize(size); zeta.resize(size + 1); zeta[0] = 1; zeta[1] = 3_M .pow((mod - 1) / size); rep(i, 2, size + 1) zeta[i] = zeta[i - 1] * zeta[1]; rep(size_log2) rep(j, (1 << i)) { ll a = (j << (size_log2 - i)) | (1 << (size_log2 - 1 - i)), b = (j + 1) << (size_log2 - i); rep(k, a, b) butterfly[k] |= 1 << i; } } void transform(vector& x) { vector ans = x; rep(size_log2) rep(j, (1 << i)) { ll zeta1 = zeta[butterfly[j * 2]], zeta2 = zeta[butterfly[j * 2 + 1]]; ll a = j << (size_log2 - i), b = (j << (size_log2 - i)) | (1 << (size_log2 - 1 - i)), c = 1 << (size_log2 - 1 - i); rep(k, a, b) { ll x1 = ans[k], x2 = ans[k | c]; ans[k] = x1 + x2 * zeta1; ans[k | c] = x1 + x2 * zeta2; } } rep(size) x[butterfly[i]] = ans[i]; } void reverse(vector& x) { vector ans = x; rep(size_log2) rep(j, (1 << i)) { ll zeta1 = zeta[size - butterfly[j * 2]], zeta2 = zeta[size - butterfly[j * 2 + 1]]; ll a = j << (size_log2 - i), b = (j << (size_log2 - i)) | (1 << (size_log2 - 1 - i)), c = 1 << (size_log2 - 1 - i); rep(k, a, b) { ll x1 = ans[k], x2 = ans[k | c]; ans[k] = x1 + x2 * zeta1; ans[k | c] = x1 + x2 * zeta2; } } rep(size) x[butterfly[i]] = ans[i] / Modint(size); } vector multiply(vector a, vector b) { a.resize(size); b.resize(size); transform(a); transform(b); rep(size) a[i] *= b[i]; reverse(a); return a; } }; signed main(){ LL(p); VEC(ll,a,p-1); VEC(ll,b,p-1); auto fac = factor(p - 1); ll g = 0; rep(i,2,p){ if(all_of(range(fac),[&i,&p](pll a){return modpow(i,(p-1)/a.first,p)!=1;})){ g=i; break; } } vec(ll,d,p-1); d[0]=1; rep(p-2)d[i+1]=d[i]*g%p; vec(Modint,A,p-1); vec(Modint,B,p-1); rep(p-1){ A[i]=a[d[i]-1]; B[i]=b[d[i]-1]; } FFT fft(p*2-3); auto C=fft.multiply(A,B); vec(Modint,c,p-1); rep(p*2-3){ c[d[i%(p-1)]-1]+=C[i]; } out(c); }