#include <vector>
#include <algorithm>
#include <cmath>
#include <queue>
#include <deque>
#include <list>
#include <unordered_map>
#include <unordered_set>
#include <iomanip>
#include <set>
#include <map>
#include <ctime>
#include <stack>
#include <functional>
#include <cstdio>
#include <string>
#include <iostream>
#include <limits>
#include <stdexcept>
#include <numeric>
#include <fstream>
#include <atcoder/all>
#include <chrono>
using namespace std;
using namespace atcoder;
using namespace internal;
typedef long long ll;
typedef long double LD;
typedef double D;
typedef pair<ll,ll> P;
typedef pair<D,D> PD;
typedef map<ll,ll> M;
#define vp(c,n) vec c(n);rep(i,n){c[i]=i;}
#define out(x) cout<<x<<endl;
#define rep(i,n) for(long long int i=0;i<n;i++)
#define rrep(i,n) for(long long int i=1;i<n;i++)
#define rep2(i, m, n) for (int i = (m); i < (n); ++i)
#define jep(i, a, n) for (ll i = (ll)a; i < (ll)n; ++i)
#define drep2(i, m, n) for (int i = (m)-1; i >= (n); --i)
#define drep(i, n) drep2(i, n, 0)
#define REP(i, n) FOR(i, 0, n)
#define fore(i,a) for(const auto &i:a)
#define RREP(var, n)  for (auto var = n - 1; var != static_cast<decltype(var)>(-1); var--)//なんですかこれは
#define all(x) (x).begin(),(x).end()
#define memset(v, h) memset((v), h, sizeof(v))
#define frac(x) x-floor(x)
#define eb emplace_back
#define pb push_back
#define pbd(x,y) push_back({x,y});
#define pbt(x,y,z) push_back({x,y,z});
#define bs binary_search 
#define lb lower_bound
#define ub upper_bound
#define bp(x) __builtin_popcount(x)
#define SUM(x) accumulate(all(x),0)
#define IP(a,b) inner_product(all(a),begin(b),0.0)//内積の計算
#define PS(a,b) partial_sum(all(a),begin(b))//累積和(具体的にはbは個数以外空にしておいて格納されていく)
#define uni(x) sort(all(x));x.erase(unique(all(x)),x.end()) //重複消去＆sort
#define fr(f,x) fixed<<setprecision(f)<<x //小数点以下f桁までの表示
#define TFU(s) transform(all(s),begin(s),::toupper);//alphabetで全て大文字にする
#define TFL(s) transform(all(s),begin(s),::tolower);//alphabetで全て小文字にする
#define replace(s,a,A) replace(all(s),'a','A')//文字列sのaをAに変換する
#define NP(a) next_permutation(all(a))//順列
#define in(a, b, x) (a<=x&&x<b)//a以上b未満
#define MAX(x) *max_element(all(x))
#define MIN(x) *min_element(all(x))
#define ROT(s,i) rotate(s.begin(),s.begin()+i,s.end())//sのi番目から後ろを前にする
#define fbit(i,n) rep(i,n)if(bit&(1<<i))//すぐ消す(見るよう)
#define foreach(y) for_each(all(y),[](ll x){cout<<x<<" ";});cout<<endl;//配列の吐き出し
#define PQ priority_queue//打ちにくいもん
#define PQD PQ<P,vector<P>,greater<P>>
#define PQS PQ<ll,vec,greater<ll>>
#define fi first//大文字になれない
#define se second
#define cauto const auto&
#define bit(n,k) ((n>>k)&1)
#define cl clock()/CLOCKS_PER_SEC//時間計測
#define printd(n,x) cout<<std::fixed<<std::setprecision(n)<<x<<endl
#define cinv(a,n); rep(i,n){cin>>a[i];}//試験運転してみる(多分つかわない)
#define cinvv(a,n,m); rep(i,n){rep(j,m){cin>>a[i][j];}}
#define popcount(n) __builtin_popcount(n)
template<class T> inline bool chmax(T& a,T b){if(a < b){a=b;return 1;}return 0;}
template<class T> inline bool chmin(T& a,T b){if(a > b){a=b;return 1;}return 0;}
typedef vector<ll> vec;
typedef vector<string> ves;
typedef vector<vec> mat;
const LD PI=acos(-1);
//const ll mod = 1000000009;//手動で切り替える.
//const ll mod = 998244353;
const ll mod = 1000000007;
//const ll mod=1e9;
//using mint =modint998244353;//.val()が必要に注意 一旦置いておく
//using mint =modint1000000009;
using mint =modint1000000007;
const ll INF = (1LL<<(60));
//ll dx[4]={0,1,0,-1},dy[4]={-1,0,1,0};
//struct Edge{ll to,co;Edge(ll to=0,ll co=0):to(to),co(co){}};//ABC191で何となくsunukeさんの！！
//struct Edge {ll to;ll cost;};
//struct Edge {ll from;ll to;ll cost;};
//struct Edge {ll to;ll w;Edge(ll to,ll w):to(to),w(w){}}; //けんちょん本 G[頂点a].pb(Edge(頂点b,weight));とか
//using Graph = vector<vector<ll>>;
//struct Edge {ll to,t,k;Edge(ll to,ll t,ll k):to(to),t(t),k(k){}};//ABC192で何となく
//using Graph =vector<vector<Edge>>;
#define cing(a,b,c) a[b].pb(c);a[c].pb(b);
//vector<bool> seen;
//void gdfs(const Graph &G,ll v){seen[v]=true;fore(next_v,G[v]){if(seen[next_v])continue;gdfs(G,next_v);}}//それぞれの点でif(seen[i])continue;else gdfs(G,i)をする
ll modpow(ll a,ll n,ll mod){if(mod==1)return 0;ll ret=1;ll p=a%mod;while(n){if(n&1)ret=ret*p%mod;p=p*p%mod;n>>=1;}return ret; }
M factor(ll n) {M ret;for(ll i=2;i*i<=n;i++){while(n%i==0){ret[i]++;n /= i;}}if(n != 1){ret[n]=1;}return ret;}//素因数分解 mapにする
vec divisor(ll n){vec K;for(ll i=1;i*i<=n;i++){if(n%i==0){K.pb(i);if(i*i!=n)K.pb(n/i);}}sort(all(K));return K;}//約数列挙 cauto& a=divisor(100);的に書く
ll modlog(ll a,ll b,ll p){ll g=1;for(ll i=p;i;i/=2)(g*=a)%=p;g=gcd(g,p);ll t=1,c=0;for(;t%g;c++){if(t==b)return c;(t*=a)%=p;}if(b%g){return -1;}t/=g;b /= g;ll n=p/g,h=0,gs=1;for(;h*h<n; h++){(gs*=a)%=n;}unordered_map<ll,ll>bs;for(ll s=0,e=b;s<h;bs[e]=++s){(e *= a) %= n;}for(ll s = 0, e = t; s < n;){(e*=gs)%=n;s+=h;if(bs.count(e))return c+s-bs[e];}return -1;}//a^x≡b(modp) x_min
bool isprime(ll N){if(N==1){return false;}if(N==2){return true;}for(ll i=2;i*i<=N;i++){if(N%i==0)return false;}return true;}
static inline ll my_div(ll n, ll p) { return double(n) / p; };
ll prime_counting(ll N) {ll N2 = sqrt(N);ll NdN2 = my_div(N, N2);vec hl(NdN2);rrep(i,NdN2){hl[i] = my_div(N, i) - 1;}vec hs(N2 + 1);iota(begin(hs), end(hs), -1);for (int x = 2, pi = 0; x <= N2; ++x) {if (hs[x] == hs[x - 1]) {continue;}ll x2 = ll(x) * x;ll imax = min<ll>(NdN2, my_div(N, x2) + 1);ll ix = x;for (ll i = 1; i < imax; ++i) {hl[i] -= (ix < NdN2 ? hl[ix] : hs[my_div(N, ix)]) - pi;ix += x;}for (int n = N2; n >= x2; n--) {hs[n] -= hs[my_div(n, x)] - pi;}++pi;}return hl[1];}

//  /*
constexpr ll MAX = 3000000;ll fac[MAX],finv[MAX],inv[MAX];//int main(){cominit();}と打つのを忘れないように.
void cominit(){fac[0]=fac[1]=1;finv[0]=finv[1]=1;inv[1]=1;for(int i=2;i<MAX;i++){fac[i]=fac[i-1]*i%mod;inv[i]=mod-inv[mod%i]*(mod/i)%mod;finv[i]=finv[i-1]*inv[i]%mod;}}
ll binom(ll n,ll k){if(n<k) return 0;if(n<0||k<0) return 0;return fac[n]*(finv[k]*finv[n-k]%mod)%mod;}
ll hom(ll n,ll k){if(n==0&&k==0) return 1;return binom(n+k-1,k);}
ll pom(ll n,ll k){if(n<k) return 0;return fac[n]*finv[n-k]%mod;}
//   */
mat binomial(ll n) {mat a(n+1,vec(n+1));rep(i,n+1)rep(j,i+1){if(j==0||j==i)a[i][j]=1;else a[i][j]=a[i-1][j-1]+a[i-1][j];}return a;}//O(n^2)
template<typename T>T phi(T n) {T ret=n;for(T i = 2; i * i <= n; i++){if(n % i == 0){ret -= ret / i;while(n % i == 0) n /= i;}}if(n > 1){ret -= ret / n;}return ret;}
template<typename T>T extgcd(T a, T b, T &x, T &y){T d = a;if(b != 0){d=extgcd(b,a%b,y,x);y -= (a / b) * x;}else {x = 1;y = 0;}return d;}
mat mat_mul(mat&a,mat&b){mat res(a.size(),vec(b[0].size()));rep(i,a.size()){rep(j,b[0].size()){rep(k,b.size()){(res[i][j]+=a[i][k]*b[k][j])%=mod;}}}return res;}
mat matpow(mat a,ll n){mat res(a.size(),vec(a.size()));rep(i,a.size())res[i][i]=1;while(n>0){if(n & 1)res=mat_mul(a,res);a=mat_mul(a,a);n>>=1;}return res;}
//#define mant vector<vector<mint>>
//mant mant_mul(mant&a,mant&b){mant res(a.size(),vector<mint>(b[0].size()));rep(i,a.size()){rep(j,b[0].size()){rep(k,b.size()){(res[i][j]+=a[i][k]*b[k][j])}}}return res;}
//mant mantpow(mant a,ll n){mant res(a.size(),vector<mint>(a.size()));rep(i,a.size())res[i][i]=1;while(n>0){if(n & 1)res=mant_mul(a,res);a=mant_mul(a,a);n>>=1;}return res;}
#define cinmat(a,x,y) mat a(x,vec(y));rep(i,x)rep(j,y)cin>>a[i][j];//;をつけなくていいよ!(お気持ちかなり使いやすそう使いにくかったらやめるけども)
#define comat(a,x,y) rep(i,x){rep(j,y){cout<<a[i][j]<<" ";}cout<<endl;}//;をつけなくていいよ！
#define FT(x) fenwick_tree<ll x> //fw(n);としてfw.add(p,x)->fw[p]+x  fw.sum(l,r)->Σ[l,r-1]fw[i] (O(logn)!!)
#define SA(s) vec suffix_arry(s)//よくわかんない
#define LCP(s,a) vec lcp_rray(s,a)//Long Common Prefix
#define ZA(s) z_algorithm(s)//上と同じようなやつ
#define IV(x,m) inv_mod(x,m)//逆元がなんと対応！！  //ついでに pair<ll,ll> crt(r,m) 解なし(0,0) n=0 (0,1) CRT!!
#define FS(n,m,a,b) floor_sum(n,m,a,b)//Σ[0,n-1]floor((a*i+b)/m) (これがなんとO(log(max(n,m,a,b))))
#define FFT(a,b,c) vec a=convolution(b,c)//FFT,NNT 畳み込み <ll m=mod>をFFT()の直後に挿入したらmodできる.
void ACC(vec a,vec &b){b[0]=0;rrep(i,a.size()){b[i]=b[i-1]+a[i-1];}return;}

struct SuccinctIndexableDictionary {size_t length;size_t blocks;vector< unsigned > bit, sum;SuccinctIndexableDictionary() = default;SuccinctIndexableDictionary(size_t length) : length(length), blocks((length + 31) >> 5) {bit.assign(blocks, 0U);sum.assign(blocks, 0U);}
void set(int k) {bit[k >> 5] |= 1U << (k & 31);}
void build() {sum[0] = 0U;for(int i = 1; i < blocks; i++) {sum[i] = sum[i - 1] + __builtin_popcount(bit[i - 1]);}}
bool operator[](int k) {return (bool((bit[k >> 5] >> (k & 31)) & 1));}
int rank(int k) {return (sum[k >> 5] + __builtin_popcount(bit[k >> 5] & ((1U << (k & 31)) - 1)));}
int rank(bool val, int k) {return (val ? rank(k) : k - rank(k));}};
template< typename T, int MAXLOG >
struct WaveletMatrix {size_t length;SuccinctIndexableDictionary matrix[MAXLOG];int mid[MAXLOG];WaveletMatrix() = default;WaveletMatrix(vector< T > v) : length(v.size()) {vector< T > l(length), r(length);
for(int level = MAXLOG - 1; level >= 0; level--) {matrix[level] = SuccinctIndexableDictionary(length + 1);int left = 0, right = 0;for(int i = 0; i < length; i++) {if(((v[i] >> level) & 1)) {matrix[level].set(i);r[right++] = v[i];} else {l[left++] = v[i];}}
mid[level] = left;matrix[level].build();v.swap(l);for(int i = 0; i < right; i++) {v[left + i] = r[i];}}}pair< int, int > succ(bool f, int l, int r, int level) {
return {matrix[level].rank(f, l) + mid[level] * f, matrix[level].rank(f, r) + mid[level] * f};}
T access(int k) {T ret = 0;for(int level = MAXLOG - 1; level >= 0; level--) {bool f = matrix[level][k];if(f) ret |= T(1) << level;k = matrix[level].rank(f, k) + mid[level] * f;}return ret;}
T operator[](const int &k) {return access(k);}int rank(const T &x, int r) {int l = 0;for(int level = MAXLOG - 1; level >= 0; level--) {tie(l, r) = succ((x >> level) & 1, l, r, level);}return r - l;}
T kth_smallest(int l, int r, int k) {assert(0 <= k && k < r - l);T ret = 0;
for(int level = MAXLOG - 1; level >= 0; level--) {int cnt = matrix[level].rank(false, r) - matrix[level].rank(false, l);bool f = cnt <= k;if(f) {ret |= T(1) << level;k -= cnt;}tie(l, r) = succ(f, l, r, level);}return ret;}
T kth_largest(int l, int r, int k) {return kth_smallest(l, r, r - l - k - 1);}
int range_freq(int l, int r, T upper) {int ret = 0;
for(int level = MAXLOG - 1; level >= 0; level--) {
bool f = ((upper >> level) & 1);if(f) {ret += matrix[level].rank(false, r) - matrix[level].rank(false, l);}tie(l, r) = succ(f, l, r, level);}return ret;}
int range_freq(int l, int r, T lower, T upper) {
return range_freq(l, r, upper) - range_freq(l, r, lower);}
T prev_value(int l, int r, T upper) {int cnt = range_freq(l, r, upper);return cnt == 0 ? T(-1) : kth_smallest(l, r, cnt - 1);}
T next_value(int l, int r, T lower) {int cnt = range_freq(l, r, lower);return cnt == r - l ? T(-1) : kth_smallest(l, r, cnt);}};
template< typename T, int MAXLOG >
struct CWM {WaveletMatrix< int, MAXLOG > mat;vector< T > ys;
CWM(const vector< T > &v) : ys(v) {sort(begin(ys), end(ys));ys.erase(unique(begin(ys), end(ys)), end(ys));vector< int > t(v.size());for(int i = 0; i < v.size(); i++) t[i] = get(v[i]);mat = WaveletMatrix< int, MAXLOG >(t);}
inline int get(const T& x) {return lower_bound(begin(ys), end(ys), x) - begin(ys);}
T access(int k) {return ys[mat.access(k)];}
T operator[](const int &k) {return access(k);}
int rank(const T &x, int r) {auto pos = get(x);if(pos == ys.size() || ys[pos] != x) {return 0;}return mat.rank(pos, r);}
T kth_smallest(int l, int r, int k) {return ys[mat.kth_smallest(l, r, k)];}
T kth_largest(int l, int r, int k) {return ys[mat.kth_largest(l, r, k)];}
int range_freq(int l, int r, T upper) {return mat.range_freq(l, r, get(upper));}
int range_freq(int l, int r, T lower, T upper) {return mat.range_freq(l, r, get(lower), get(upper));}
T prev_value(int l, int r, T upper) {auto ret = mat.prev_value(l, r, get(upper));return ret == -1 ? T(-1) : ys[ret];}
T next_value(int l, int r, T lower) {auto ret = mat.next_value(l, r, get(lower));return ret == -1 ? T(-1) : ys[ret];}};
void cum(vec &a,vec b){
ll n=b.size();
a.resize(n);
rep(i,n-1){a[i+1]=a[i]+b[i];}
}
void cum1(vec &a,vec b){
ll n=b.size();
a.resize(n-1);
a[0]=b[0];
rrep(i,n-1){a[i]=a[i-1]+b[i];}
}





namespace inner {
 
using i32 = int32_t;
using u32 = uint32_t;
using i64 = int64_t;
using u64 = uint64_t;
 
template <typename T>
T gcd(T a, T b) {
  while (b) swap(a %= b, b);
  return a;
}
 
template <typename T>
T inv(T a, T p) {
  T b = p, x = 1, y = 0;
  while (a) {
    T q = b / a;
    swap(a, b %= a);
    swap(x, y -= q * x);
  }
  assert(b == 1);
  return y < 0 ? y + p : y;
}
 
template <typename T, typename U>
T modpow(T a, U n, T p) {
  T ret = 1 % p;
  for (; n; n >>= 1, a = U(a) * a % p)
    if (n & 1) ret = U(ret) * a % p;
  return ret;
}
 
}  // namespace inner
namespace my_rand {
 
// [0, 2^64 - 1)
uint64_t rng() {
  static uint64_t x_ =
      uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                   chrono::high_resolution_clock::now().time_since_epoch())
                   .count()) *
      10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}
 
// [l, r)
int64_t randint(int64_t l, int64_t r) {
  assert(l < r);
  return l + rng() % (r - l);
}
 
// choose n numbers from [l, r) without overlapping
vector<int64_t> randset(int64_t l, int64_t r, int64_t n) {
  assert(l <= r && n <= r - l);
  unordered_set<int64_t> s;
  for (int64_t i = n; i; --i) {
    int64_t m = randint(l, r + 1 - i);
    if (s.find(m) != s.end()) m = r - i;
    s.insert(m);
  }
  vector<int64_t> ret;
  for (auto& x : s) ret.push_back(x);
  return ret;
}
 
// [0.0, 1.0)
double rnd() {
  union raw_cast {
    double t;
    uint64_t u;
  };
  constexpr uint64_t p = uint64_t(1023 - 64) << 52;
  return rng() * ((raw_cast*)(&p))->t;
}
 
template <typename T>
void randshf(vector<T>& v) {
  int n = v.size();
  for (int loop = 0; loop < 2; loop++)
    for (int i = 0; i < n; i++) swap(v[i], v[randint(0, n)]);
}
 
}  // namespace my_rand
using my_rand::randint;using my_rand::randset;using my_rand::randshf;using my_rand::rnd;using my_rand::rng;
struct ArbitraryLazyMontgomeryModInt {
  using mint = ArbitraryLazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;
 
  static u32 mod;
  static u32 r;
  static u32 n2;
 
  static u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }
 
  static void set_mod(u32 m) {
    assert(m < (1 << 30));
    assert((m & 1) == 1);
    mod = m;
    n2 = -u64(m) % m;
    r = get_r();
    assert(r * mod == 1);
  }
 
  u32 a;
 
  ArbitraryLazyMontgomeryModInt() : a(0) {}
  ArbitraryLazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};
 
  static u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }
 
  mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }
 
  mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }
 
  mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }
 
  mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }
 
  mint operator+(const mint &b) const { return mint(*this) += b; }
  mint operator-(const mint &b) const { return mint(*this) -= b; }
  mint operator*(const mint &b) const { return mint(*this) *= b; }
  mint operator/(const mint &b) const { return mint(*this) /= b; }
  bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  mint operator-() const { return mint() - mint(*this); }
 
  mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
 
  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }
 
  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = ArbitraryLazyMontgomeryModInt(t);
    return (is);
  }
 
  mint inverse() const { return pow(mod - 2); }
 
  u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }
 
  static u32 get_mod() { return mod; }
};
typename ArbitraryLazyMontgomeryModInt::u32 ArbitraryLazyMontgomeryModInt::mod;typename ArbitraryLazyMontgomeryModInt::u32 ArbitraryLazyMontgomeryModInt::r;typename ArbitraryLazyMontgomeryModInt::u32 ArbitraryLazyMontgomeryModInt::n2;
struct montgomery64 {
  using mint = montgomery64;
  using i64 = int64_t;
  using u64 = uint64_t;
  using u128 = __uint128_t;
 
  static u64 mod;
  static u64 r;
  static u64 n2;
 
  static u64 get_r() {
    u64 ret = mod;
    for (i64 i = 0; i < 5; ++i) ret *= 2 - mod * ret;
    return ret;
  }
 
  static void set_mod(u64 m) {
    assert(m < (1LL << 62));
    assert((m & 1) == 1);
    mod = m;
    n2 = -u128(m) % m;
    r = get_r();
    assert(r * mod == 1);
  }
 
  u64 a;
 
  montgomery64() : a(0) {}
  montgomery64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){};
 
  static u64 reduce(const u128 &b) {
    return (b + u128(u64(b) * u64(-r)) * mod) >> 64;
  }
 
  mint &operator+=(const mint &b) {
    if (i64(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }
 
  mint &operator-=(const mint &b) {
    if (i64(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }
 
  mint &operator*=(const mint &b) {
    a = reduce(u128(a) * b.a);
    return *this;
  }
 
  mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }
 
  mint operator+(const mint &b) const { return mint(*this) += b; }
  mint operator-(const mint &b) const { return mint(*this) -= b; }
  mint operator*(const mint &b) const { return mint(*this) *= b; }
  mint operator/(const mint &b) const { return mint(*this) /= b; }
  bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  mint operator-() const { return mint() - mint(*this); }
 
  mint pow(u128 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
 
  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }
 
  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = montgomery64(t);
    return (is);
  }
 
  mint inverse() const { return pow(mod - 2); }
 
  u64 get() const {
    u64 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }
 
  static u64 get_mod() { return mod; }
};
typename montgomery64::u64 montgomery64::mod, montgomery64::r, montgomery64::n2;
namespace fast_factorize {
using u64 = uint64_t;
template <typename mint>
bool miller_rabin(u64 n, vector<u64> as) {
  if (mint::get_mod() != n) mint::set_mod(n);
  u64 d = n - 1;
  while (~d & 1) d >>= 1;
  mint e{1}, rev{int64_t(n - 1)};
  for (u64 a : as) {
    if (n <= a) break;
    u64 t = d;
    mint y = mint(a).pow(t);
    while (t != n - 1 && y != e && y != rev) {
      y *= y;
      t *= 2;
    }
    if (y != rev && t % 2 == 0) return false;
  }
  return true;
}
 
bool is_prime(u64 n) {
  if (~n & 1) return n == 2;
  if (n <= 1) return false;
  if (n < (1LL << 30))
    return miller_rabin<ArbitraryLazyMontgomeryModInt>(n, {2, 7, 61});
  else
    return miller_rabin<montgomery64>(
        n, {2, 325, 9375, 28178, 450775, 9780504, 1795265022});
}
 
template <typename mint, typename T>
T pollard_rho(T n) {
  if (~n & 1) return 2;
  if (is_prime(n)) return n;
  if (mint::get_mod() != n) mint::set_mod(n);
  mint R, one = 1;
  auto f = [&](mint x) { return x * x + R; };
  auto rnd_ = [&]() { return rng() % (n - 2) + 2; };
  while (1) {
    mint x, y, ys, q = one;
    R = rnd_(), y = rnd_();
    T g = 1;
    constexpr int m = 128;
    for (int r = 1; g == 1; r <<= 1) {
      x = y;
      for (int i = 0; i < r; ++i) y = f(y);
      for (int k = 0; g == 1 && k < r; k += m) {
        ys = y;
        for (int i = 0; i < m && i < r - k; ++i) q *= x - (y = f(y));
        g = inner::gcd<T>(q.get(), n);
      }
    }
    if (g == n) do
        g = inner::gcd<T>((x - (ys = f(ys))).get(), n);
      while (g == 1);
    if (g != n) return g;
  }
  exit(1);
}
 
using i64 = long long;
 
vector<i64> inner_factorize(u64 n) {
  if (n <= 1) return {};
  u64 p;
  if (n <= (1LL << 30))
    p = pollard_rho<ArbitraryLazyMontgomeryModInt, uint32_t>(n);
  else
    p = pollard_rho<montgomery64, uint64_t>(n);
  if (p == n) return {i64(p)};
  auto l = inner_factorize(p);
  auto r = inner_factorize(n / p);
  copy(begin(r), end(r), back_inserter(l));
  return l;
}
 
vector<i64> factorize(u64 n) {
  auto ret = inner_factorize(n);
  sort(begin(ret), end(ret));
  return ret;
}
 
map<i64, i64> factor_count(u64 n) {
  map<i64, i64> mp;
  for (auto &x : factorize(n)) mp[x]++;
  return mp;
}
 
vector<i64> divisors(u64 n) {
  if (n == 0) return {};
  vector<pair<i64, i64>> v;
  for (auto &p : factorize(n)) {
    if (v.empty() || v.back().first != p) {
      v.emplace_back(p, 1);
    } else {
      v.back().second++;
    }
  }
  vector<i64> ret;
  auto f = [&](auto rc, int i, i64 x) -> void {
    if (i == (int)v.size()) {
      ret.push_back(x);
      return;
    }
    for (int j = v[i].second;; --j) {
      rc(rc, i + 1, x);
      if (j == 0) break;
      x *= v[i].first;
    }
  };
  f(f, 0, 1);
  sort(begin(ret), end(ret));
  return ret;
}
 
}  // namespace fast_factorize
using fast_factorize::divisors;using fast_factorize::factor_count;using fast_factorize::factorize;using fast_factorize::is_prime;
using namespace fast_factorize;
vec base(ll x,ll b) {vec ret;ll t = 1, k = abs(b);
while(x) {ret.emplace_back((x * t) % k);if(ret.back() < 0){ret.back() += k;}x -= ret.back() * t;x /= k;t *= b / k;}
if(ret.empty()) {ret.emplace_back(0);}reverse(begin(ret), end(ret));return ret;}
 
 
 
 
 
struct Barrett {
  using u32 = unsigned int;
  using i64 = long long;
  using u64 = unsigned long long;
  u32 m;
  u64 im;
  Barrett() : m(), im() {}
  Barrett(int n) : m(n), im(u64(-1) / m + 1) {}
  constexpr inline i64 quo(u64 n) {
    u64 x = u64((__uint128_t(n) * im) >> 64);
    u32 r = n - x * m;
    return m <= r ? x - 1 : x;
  }
  constexpr inline i64 rem(u64 n) {
    u64 x = u64((__uint128_t(n) * im) >> 64);
    u32 r = n - x * m;
    return m <= r ? r + m : r;
  }
  constexpr inline pair<i64, int> quorem(u64 n) {
    u64 x = u64((__uint128_t(n) * im) >> 64);
    u32 r = n - x * m;
    if (m <= r) return {x - 1, r + m};
    return {x, r};
  }
  constexpr inline i64 pow(u64 n, i64 p) {
    u32 a = rem(n), r = m == 1 ? 0 : 1;
    while (p) {
      if (p & 1) r = rem(u64(r) * a);
      a = rem(u64(a) * a);
      p >>= 1;
    }
    return r;
  }
};
struct prime_power_binomial {
  int p, q, M;
  vector<int> fac, ifac, inv;
  int delta;
  Barrett bm, bp;
 
  prime_power_binomial(int _p, int _q) : p(_p), q(_q) {
    assert(1 < p && p <= ((1LL << 30) - 1));
    assert(_q > 0);
    long long m = 1;
    while (_q--) {
      m *= p;
      assert(m <=((1LL << 30) - 1));
    }
    M = m;
    bm = Barrett(M), bp = Barrett(p);
    enumerate();
    delta = (p == 2 && q >= 3) ? 1 : M - 1;
  }
 
  void enumerate() {
    int MX = min<int>(M, 20000000 + 10);
    fac.resize(MX);
    ifac.resize(MX);
    inv.resize(MX);
    fac[0] = ifac[0] = inv[0] = 1;
    fac[1] = ifac[1] = inv[1] = 1;
    for (int i = 2; i < MX; i++) {
      if (i % p == 0) {
        fac[i] = fac[i - 1];
        fac[i + 1] = bm.rem(1LL * fac[i - 1] * (i + 1));
        i++;
      } else {
        fac[i] = bm.rem(1LL * fac[i - 1] * i);
      }
    }
    ifac[MX - 1] = bm.pow(fac[MX - 1], M / p * (p - 1) - 1);
    for (int i = MX - 2; i > 1; --i) {
      if (i % p == 0) {
        ifac[i] = bm.rem(1LL * ifac[i + 1] * (i + 1));
        ifac[i - 1] = ifac[i];
        i--;
      } else {
        ifac[i] = bm.rem(1LL * ifac[i + 1] * (i + 1));
      }
    }
  }
 
  long long Lucas(long long n, long long m) {
    int res = 1;
    while (n) {
      int n0, m0;
      tie(n, n0) = bp.quorem(n);
      tie(m, m0) = bp.quorem(m);
      if (n0 < m0) return 0;
      res = bm.rem(1LL * res * fac[n0]);
      int buf = bm.rem(1LL * ifac[n0 - m0] * ifac[m0]);
      res = bm.rem(1LL * res * buf);
    }
    return res;
  }
 
  long long C(long long n, long long m) {
    if (n < m || n < 0 || m < 0) return 0;
    if (q == 1) return Lucas(n, m);
    long long r = n - m;
    int e0 = 0, eq = 0, i = 0;
    int res = 1;
    while (n) {
      res = bm.rem(1LL * res * fac[bm.rem(n)]);
      res = bm.rem(1LL * res * ifac[bm.rem(m)]);
      res = bm.rem(1LL * res * ifac[bm.rem(r)]);
      n = bp.quo(n);
      m = bp.quo(m);
      r = bp.quo(r);
      int eps = n - m - r;
      e0 += eps;
      if (e0 >= q) return 0;
      if (++i >= q) eq += eps;
    }
    if (eq & 1) res = bm.rem(1LL * res * delta);
    res = bm.rem(1LL * res * bm.pow(p, e0));
    return res;
  }
};
struct binomial_mod {
  int mod;
  vector<int> M;
  vector<prime_power_binomial> cs;
 
  binomial_mod(long long md) : mod(md) {
    assert(1 <= md);
    assert(md <= ((1LL << 30) - 1));
    for (int i = 2; i * i <= md; i++) {
      if (md % i == 0) {
        int j = 0, k = 1;
        while (md % i == 0) md /= i, j++, k *= i;
        M.push_back(k);
        cs.emplace_back(i, j);
        assert(M.back() == cs.back().M);
      }
    }
    if (md != 1) {
      M.push_back(md);
      cs.emplace_back(md, 1);
    }
    assert(M.size() == cs.size());
  }
 
  long long binom(long long n, long long m) {
    if (mod == 1) return 0;
    vector<long long> rem, d;
    for (int i = 0; i < (int)cs.size(); i++) {
      rem.push_back(cs[i].C(n, m));
      d.push_back(M[i]);
    }
    return atcoder::crt(rem, d).first;
  }
};

#define ip(x) is_prime_constexpr(x)



int main(){
ll n,k,p;
cin>>n>>k>>p;
vec a(n),b(n);
cinv(a,n);
cinv(b,n);
unordered_map<ll,ll> mp;
rep(i,n){
mp[b[i]]++;
mp[b[i]+p]++;
}
unordered_map<ll,ll> c;
c[0]=mp[0];
rep(i,2*p+1){c[i+1]=c[i]+mp[i+1];}
auto f=[&](ll x){
ll res=0;
rep(i,n){
res+=c[p-a[i]+x-1]-c[p-a[i]-1];
}
if(res<k){return true;}
else{return false;}
};
ll le=0,ri=p;
ll ans=INF;
while(ri-le>1){
ll mi=(le+ri)/2;
if(f(mi)){le=mi;}
else{ri=mi;chmin(ans,mi);}
}
cout<<le<<endl;
}