#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;
template <class T> using VV = V<V<T>>;
template <class T> using VVV = V<VV<T>>;
template <class S, class T> using P = pair<S, T>;
template <class S, class T, class U> using TP = tuple<S, T, U>;

using ll = long long;
using ull = unsigned long long;
using dbl = double;
using str = string;
using vll = V<ll>;
using vvll = V<vll>;
using vvvll = V<vvll>;
using pl = P<ll, ll>;
using tl = TP<ll, ll, ll>;
using vpl = V<pl>;
using vvpl = V<vpl>;
using vtl = V<tl>;
using vvtl = V<vtl>;
using vs = V<str>;
using vvs = V<vs>;
using vd = V<dbl>;
using vvd = V<vd>;
using qll = queue<ll>;
using qpl = queue<pl>;
using stll = stack<ll>;
using stpl = stack<pl>;
using mapll = map<ll, ll>;
using setll = set<ll>;
using pqll = priority_queue<ll>;

#define int ll
#define fi first
#define se second
#define mp make_pair
#define mt make_tuple
#define pb push_back
#define pob pop_back()
#define pf push_front
#define pof pop_front()
#define sz size()
#define bgn begin()
#define en end()
#define asn assign
#define emp empty()
#define fr front()
#define bk back()
#define clr clear()
#define ins insert
#define ers erase
#define res resize
#define tp top()
#define p_q priority_queue
#define inv inverse()

#define FOR(i,a,b) for(ll i=(a);i<=(ll)(b);i++)
#define rFOR(i,a,b) for(ll i=(b);i>=(ll)(a);i--)
#define REP(i,a) FOR((i),0,(ll)(a)-1)
#define REP0(i,a) FOR((i),0,(ll)(a))
#define REP1(i,a) FOR((i),1,(ll)(a))
#define rREP(i,a) rFOR((i),0,(ll)(a)-1)
#define rREP0(i,a) rFOR((i),0,(ll)(a))
#define rREP1(i,a) rFOR((i),1,(ll)(a))
#define ROR(v,i) for(auto &(i):(v))
#define IOTA(a,n) iota((a).bgn,(a).en,(n))
#define SORT(a) sort((a).bgn,(a).en)
#define rSORT(a) sort((a).rbegin(),(a).rend())
#define UNIQUE(a) (a).erase(unique((a).bgn,(a).en),(a).en)
#define PREVP(a) prev_permutation((a).bgn,(a).en)
#define NEXTP(a) next_permutation((a).bgn,(a).en)
#define BINS(a,b) binary_search((a).bgn,(a).en,(b))
#define LOWB(a,b) (lower_bound((a).bgn,(a).en,(b))-(a).bgn)
#define UPB(a,b) (upper_bound((a).bgn,(a).en,(b))-(a).bgn)
#define CNT(a,b) count((a).bgn,(a).en,b)
#define SUM(a) accumulate((a).bgn,(a).en,0)
#define REV(a) reverse((a).bgn,(a).en)
#define REGS(a,b) regex_search((a),regex(b))
#define REGM(a,b) regex_match((a),regex(b))
#define yn(a) cout <<((a)?"yes":"no")<<"\n";
#define Yn(a) cout <<((a)?"Yes":"No")<<"\n";
#define YN(a) cout <<((a)?"YES":"NO")<<"\n";
#define imp(a) cout <<((a)?"possible":"impossible")<<"\n";
#define Imp(a) cout <<((a)?"Possible":"Impossible")<<"\n";
#define IMP(a) cout <<((a)?"POSSIBLE":"IMPOSSIBLE")<<"\n";
#define fs(a) cout <<((a)?"second":"first")<<"\n";
#define Fs(a) cout <<((a)?"Second":"First")<<"\n";
#define FS(a) cout <<((a)?"SECOND":"FIRST")<<"\n";
#define sak cout <<"\n";
#define sas cout <<" ";
#define sat cout <<"\t";
#define dbg(a) cerr <<(#a)<<": "<<(a)<<"\n";
#define dbgs(...) dal(#__VA_ARGS__);dal(__VA_ARGS__);
#define c2l(a) ((ll)(a-48))
#define a2l(a) ((ll)(a-97))
#define A2l(a) ((ll)(a-65))
#define l2c(a) ((char)(a+48))
#define l2a(a) ((char)(a+97))
#define l2A(a) ((char)(a+65))
#define DigN2(a) ((llabs(a)==0)?(1):((ll)(log2(double(llabs(a))))+1))
#define DigN10(a) ((llabs(a)==0)?(1):((ll)(log10(double(llabs(a))))+1))
#define Dig2(a,b) (((a)>>(b))&1)
#define Dig10(a,b) (ll)(((a)/((ll)(pow(10.0,(double)(b)))))%10)
#define Pow2(a) ((ll)(1)<<(a))
#define Pow10(a) ((ll)(pow(10.0,double(a))))
#define LSB(a) ((a)&(-(a)))
#define vin(v) ROR((v),(i)){cin >>(i);};
#define vllin(v,N) vll (v)((N));vin(v);
#define vllin2(a,b,N) vll (a)(N),(b)(N);REP(i,N){cin>>(a)[i]>>(b)[i];};
#define vsin(v,N) vs (v)((N));vin(v);
#define rdn(a,b) ((a)/(b))
#define rou(a,b) ((((double(a)/double(b))-((a)/(b)))<0.5)?((a)/(b)):(((a)/(b))+1))
#define rup(a,b) ((((a)%(b))==0)?((a)/(b)):(((a)/(b))+1))
#define min(...) Min(__VA_ARGS__)
#define max(...) Max(__VA_ARGS__)
#define emin(a, ...) ((a)=Min((a),__VA_ARGS__))
#define emax(a, ...) ((a)=Max((a),__VA_ARGS__))
#define egcd(a, ...) ((a)=gcd((a),__VA_ARGS__))
#define elcm(a, ...) ((a)=lcm((a),__VA_ARGS__))
#define powll(a,b) (ll)(pow((double)(a),(double)(b)))
#define Triangle(x1,y1,x2,y2,x3,y3) (((x1)-(x2))*((y1)-(y3))-((x1)-(x3))*((y1)-(y2)))

#define svll SumV<ll>
#define svvll SumV2<ll>

#define li(...) ll __VA_ARGS__;Input(__VA_ARGS__);
#define si(...) str __VA_ARGS__;Input(__VA_ARGS__);
#define vli(size, ...) vll __VA_ARGS__;vInput(size,__VA_ARGS__);
#define vsi(size, ...) vs __VA_ARGS__;vInput(size,__VA_ARGS__);

//const ll MOD = 1e9 + 7;
const ll MOD = 998244353;
//const ll MOD = 924844033;
//const ll MOD = 9007199254740881;
const ll INF = 1LL << 60;//1.15e18
const double PI = acos(-1.0);
const vll DX = { 0,-1,0,1,0,-1,1,1,-1 };
const vll DY = { 0,0,-1,0,1,-1,-1,1,1 };
const str alp = "abcdefghijklmnopqrstuvwxyz";
const str ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";


void Input() {}
template <class Var, class... Args> void Input(Var& var, Args&... args) {
    cin >> var;
    Input(args...);
}

void vInit(ll size) {}
template <class T, class... Args> void vInit(ll size, V<T>& v, Args&... args) {
    v.res(size);
    vInit(size, args...);
}
void vInputNum(ll num) {}
template <class T, class... Args> void vInputNum(ll num, V<T>& v, Args&... args) {
    cin >> v[num];
    vInputNum(num, args...);
}
void vInput(ll size) {}
template <class... Args> void vInput(ll size, Args&... args) {
    vInit(size, args...);
    REP(i, size) vInputNum(i, args...);
}

template <class S, class T> ostream &operator<<(ostream &out, const P<S, T> &p) {
    return out << "[" << p.fi << ", " << p.se << "]";
}

template <class T> ostream &operator<<(ostream &out, V<T> &v) {
    if (v.emp) return out << "{}";
    else {
        auto itr = v.bgn;
        out << "{" << *itr;
        itr++;
        while (itr != v.en) {
            out << ", " << *itr;
            itr++;
        }
        out << "}";
        return out;
    }
}

template <class S, class T> ostream &operator<<(ostream &out, const map<S, T> &m) {
    if (m.emp) return out << "<[]>";
    else {
        auto itr = m.bgn;
        out << "< [" << (itr->fi) << ": " << (itr->se);
        itr++;
        while (itr != m.en) {
            out << "], [" << (itr->fi) << ": " << (itr->se);
            itr++;
        }
        out << "] >";
        return out;
    }
}

template <class T> ostream &operator<<(ostream &out, const set<T> &s) {
    if (s.emp) return out << "<>";
    else {
        auto itr = s.bgn;
        out << "<" << *itr;
        itr++;
        while (itr != s.en) {
            out << ", " << *itr;
            itr++;
        }
        out << ">";
        return out;
    }
}

void say() {}
template <class T> void say(T t) { cout << t; }
template <class Head, class... Body> void say(Head head, Body... body) {
    cout << head << " ";
    say(body...);
}
void sal() { cout << "\n"; }
template <class... Args> void sal(Args... args) {
    say(args...);
    cout << "\n";
}

void day() {}
template <class T> void day(T t) { cerr << t; }
template <class Head, class... Body> void day(Head head, Body... body) {
    cerr << head << " ";
    day(body...);
}
void dal() { cerr << "\n"; }
template <class... Args> void dal(Args... args) {
    day(args...);
    cerr << "\n";
}

void salv() {}
template <class T> void salv(V<T> v) {
    if (v.emp) sal();
    else {
        auto itr = v.bgn;
        say(*itr);
        itr++;
        while (itr != v.en) {
            sas;
            say(*itr);
            itr++;
        }
        sak;
    }
}
template <class T, class... Args> void salv(V<T> v, Args... args) {
    salv(v);
    salv(args...);
}

void salv2() {}
template <class T> void salv2(V<V<T>> v) {
    if (v.emp) sal();
    else {
        ROR(v, i) salv(i);
    }
}
template <class T, class... Args> void salv2(V<V<T>> v, Args... args) {
    salv2(v);
    salv2(args...);
}

template <class Monoid> struct Action {
public:
    Monoid I;
    function<Monoid(Monoid, Monoid)> A;
    Action(Monoid I, function<Monoid(Monoid, Monoid)> A) : I(I), A(A) {}
    Monoid operator()() { return I; }
    Monoid operator()(Monoid x) { return x; }
    Monoid operator()(Monoid l, Monoid r) { return A(l, r); }
    template <class... Args> Monoid operator()(Monoid x, Args... args) {
        Monoid tmp = operator()(args...);   
        return A(x, tmp);
    }
};

template <class T> Action<T> ADD = Action<T>(0, [](T l, T r) { return l + r; });
template <> Action<str> ADD<str> = Action<str>("", [](str l, str r) { return l + r; });
template <class T1, class T2> Action<P<T1, T2>> ADD<P<T1, T2>> = Action<P<T1, T2>>(mp(ADD<T1>.I, ADD<T2>.I), [](P<T1, T2> l, P<T1, T2> r) { return mp(l.fi + r.fi, l.se + r.se); });

template <class T> Action<T> MUL = Action<T>(1, [](T l, T r) { return l * r; });
template <class T> Action<T> OR = Action<T>(0, [](T l, T r) { return l | r; });
template <class T> Action<T> XOR = Action<T>(0, [](T l, T r) { return l ^ r; });

template <class T> Action<T> AND = Action<T>(((ll)(1) << 63) - 1, [](T l, T r) { return l & r; });
template <> Action<bool> AND<bool> = Action<bool>(true, [](bool l, bool r) { return l & r; });
template <> Action<ull> AND<ull> = Action<ull>(((ull)(1) << 63) - 1, [](ull l, ull r) { return l & r; });

template <class T> Action<T> MIN = Action<T>(INF, [](T l, T r) { return (l < r) ? l : r; });
template <class T> Action<T> MAX = Action<T>(-INF, [](T l, T r) { return (l > r) ? l : r; });

template <class T> Action<T> GCD = Action<T>(0, [](T l, T r) { if (l < r) { l ^= r; r ^= l; l ^= r; } return (r ? GCD<T>(r, l%r) : l); });
template <class T> Action<T> LCM = Action<T>(1, [](T l, T r) { return (l * r) / GCD<T>(l, r); });


template <class Head> Head Min(Head head) { return head; }
template <class Head, class... Body> Head Min(Head head, Body... body) {
    auto tmp = Min(body...);
    return (head < tmp) ? head : tmp;
}

template <class Head> Head Max(Head head) { return head; }
template <class Head, class... Body> auto Max(Head head, Body... body) {
    auto tmp = Max(body...);
    return (head > tmp) ? head : tmp;
}

ll gcd(ll a, ll b) { if (a < b) { a ^= b; b ^= a; a ^= b; } return b ? gcd(b, a%b) : a; }
ll lcm(ll a, ll b) { return a / gcd(a, b)*b; }

ll gcd(ll head) { return head; }
template <class... Body> ll gcd(ll head, Body... body) {
    return gcd(head, gcd(body...));
}

ll lcm(ll head) { return head; }
template <class... Body> ll lcm(ll head, Body... body) {
    return lcm(head, lcm(body...));
}


ll Bset(ll a, ll b, ll c) {
    if (c) a |= b;
    else a &= ~b;
    return a;
}


struct UFT {

public:
    ll tsize;
    ll mode;
    vll par;
    vll rank;
    UFT() {}
    UFT(const UFT &uft) {}
    UFT(ll tsizeget, ll modeget = 0){
        tsize = tsizeget;
        mode = modeget;
        par.asn(tsize, -1);
        if (!mode) rank.res(tsize, 0);
    }
    ll root(ll x) {
        return par[x] < 0 ? x : par[x] = root(par[x]);
    }
    bool isRoot(ll x) {
        return x == root(x);
    }
    bool same(ll x, ll y) {
        return root(x) == root(y);
    }
    void merge(ll x, ll y) {
        x = root(x);
        y = root(y);
        if (x == y) return;
        if (mode) {
            par[x] += par[y];
            par[y] = x;
        }
        else {
            if (rank[x] < rank[y]) {
                par[y] += par[x];
                par[x] = y;
            }
            else {
                par[x] += par[y];
                par[y] = x;
                if (rank[x] == rank[y]) rank[x]++;
            }
        }
    }
    ll size(ll x) {
        return -par[root(x)];
    }
};

struct pUFT {

public:

    ll tsize;
    ll now;
    vll par;
    vll rank;
    vll mtime;
    vvll sizepi;
    vvll sizepv;

    pUFT(){}
    pUFT(const pUFT &puft) {}

    pUFT(ll tsizeget) {
        tsize = tsizeget;
        now = 0;
        par.asn(tsize, -1);
        rank.asn(tsize, 0);
        mtime.asn(tsize, INF);
        sizepi.asn(tsize, { 0 });
        sizepv.asn(tsize, { 1 });
    }
    ll root(ll x, ll t) {
        return (mtime[x] > t) ? x : root(par[x], t);
    }
    bool same(ll x, ll y, ll t) {
        return root(x, t) == root(y, t);
    }
    ll merge(ll x, ll y) {
        now++;
        x = root(x, now);
        y = root(y, now);
        if (x != y) {
            if (rank[x] < rank[y]) {
                par[y] += par[x];
                sizepi[y].pb(now);
                sizepv[y].pb(-par[y]);
                par[x] = y;
                mtime[x] = now;
            }
            else {
                par[x] += par[y];
                sizepi[x].pb(now);
                sizepv[x].pb(-par[x]);
                par[y] = x;
                mtime[y] = now;
                if (rank[x] == rank[y]) rank[x]++;
            }
        }
        return now;
    }
    ll size(ll x, ll t) {
        x = root(x, t);
        return sizepv[x][UPB(sizepi[x], t) - 1];
    }

};

struct wUFT {
public:
    ll tsize;
    ll mode;
    vll par;
    vll rank;
    vll dweight;
    wUFT() {}
    wUFT(const wUFT &wuft) {}
    wUFT(ll tsizeget, ll modeget = 0) {
        tsize = tsizeget;
        mode = modeget;
        par.asn(tsize, -1);
        if (!mode) rank.res(tsize, 0);
        dweight.asn(tsize, 0);
    }
    ll root(ll x) {
        if (par[x] < 0) return x;
        else {
            ll r = root(par[x]);
            dweight[x] += dweight[par[x]];
            return par[x] = r;
        }
    }
    ll weight(ll x) {
        root(x);
        return dweight[x];
    }
    ll diff(ll x, ll y) {
        return weight(y) - weight(x);
    }
    bool isRoot(ll x) {
        return x == root(x);
    }
    bool same(ll x, ll y) {
        return root(x) == root(y);
    }
    void merge(ll x, ll y, ll w) {
        w += weight(x);
        w -= weight(y);
        x = root(x);
        y = root(y);
        if (x == y) return;
        if (mode) {
            par[x] += par[y];
            par[y] = x;
        }
        else {
            if (rank[x] < rank[y]) {
                par[y] += par[x];
                par[x] = y;
                dweight[x] = -w;
            }
            else {
                par[x] += par[y];
                par[y] = x;
                if (rank[x] == rank[y]) rank[x]++;
                dweight[y] = w;
            }
        }
    }
    ll size(ll x) {
        return -par[root(x)];
    }
};

template <typename valtype> class SegT {

public:

    ll size;
    vector<valtype> v;

    valtype initv;
    function<valtype(valtype x, valtype y)> calc;

    SegT() {}
    SegT(const SegT &segt) {}

    SegT(ll sizeget, ll modeget = 0) {
        sizeset(sizeget);
        modeset(modeget);
        init();
    }
    SegT(vector<valtype> cpyvec, ll modeget = 0) {
        sizeset(cpyvec.sz);
        modeset(modeget);
        init();
        copy(cpyvec);
    }
    SegT(ll sizeget, valtype initvget, function<valtype(valtype x, valtype y)> calcget) {
        sizeset(sizeget);
        initv = initvget;
        calc = calcget;
        init();
    }
    SegT(vector<valtype> cpyvec, valtype initvget, function<valtype(valtype x, valtype y)> calcget) {
        sizeset(cpyvec.sz);
        initv = initvget;
        calc = calcget;
        init();
        copy(cpyvec);
    }
    void sizeset(ll rsize) {
        size = DigN2(rsize);
        if (rsize == Pow2(size - 1)) size--;
        return;
    }
    void modeset(ll mode) {
        switch (mode) {
        case 0:
            initv = 0;
            calc = [](valtype x, valtype y) {return x + y; };
            break;
        case 1:
            initv = INF;
            calc = [](valtype x, valtype y) {return min(x, y); };
            break;
        case 2:
            initv = -INF;
            calc = [](valtype x, valtype y) {return max(x, y); };
            break;
        }
        return;
    }
    void init() {
        v.asn(Pow2(size + 1) - 1, initv);
    }
    void copy(vector<valtype> cpyvec) {
        REP(i, min(cpyvec.sz, Pow2(size))) set(i, cpyvec[i]);
    }
    ll i2v(ll i) const{
        if (i < 0 || i >= Pow2(size)) return -1;
        return Pow2(size) + i - 1;
    }
    ll top(ll i) const{
        if (i == 0) return -1;
        return (i - 1) / 2;
    }
    pl bot(ll i) const{
        if (i + 1 >= Pow2(size)) return mp(-1, -1);
        return mp(2 * i + 1, 2 * i + 2);
    }
    void set(ll i, valtype x) {
        i = i2v(i);
        v[i] = x;
        while (i > 0) {
            i = top(i);
            v[i] = calc(v[bot(i).fi], v[bot(i).se]);
        }
        return;
    }
    void add(ll i, valtype x) {
        set(i, v[i2v(i)] + x);
        return;
    }
    valtype operator[](const ll &i) const {
        return v[i2v(i)];
    }
//  valtype que(ll a = 0, ll b = Pow2(size) - 1) {
    valtype que(ll a, ll b) {
        if (a == b) return v[i2v(a)];
        if (a > b) return initv;//swap(a, b);
        valtype ans = initv;
        ll ai = i2v(a);
        ll bi = i2v(b);
        FOR(i, 1, size + 1) {
            if (a > b) break;
            if (a%Pow2(i)) {
                ans = calc(ans, v[ai]);
                a += Pow2(i - 1);
                ai = top(ai) + 1;
            }
            else {
                ai = top(ai);
            }
            if (a > b) break;
            if ((b + 1) % Pow2(i)) {
                ans = calc(ans, v[bi]);
                b -= Pow2(i - 1);
                bi = top(bi) - 1;
            }
            else {
                bi = top(bi);
            }
            if (a > b) break;
        }
        return ans;
    }
    valtype que(ll b) {
        return que(0, b);
    }
    valtype que() {
        return que(0, Pow2(size) - 1);
    }


};

template <typename lentype> class Graph {

public:
    
    ll size;
    ll mode;
    ll mapmode;
    lentype zlen;
    lentype ilen;
    vector<map<ll, ll>> v2n;
    vvll n2v;
    vector<vector<lentype>> Edge;
    vector<pair<lentype, ll>> Primresult;

    Graph() {}
    Graph(const Graph &graph) {}

    Graph(ll sizeget = 0, ll mapmodeget = 0, ll modeget = 0, lentype zlenget = 0, lentype ilenget = INF) {
        size = sizeget;
        mapmode = mapmodeget;
        mode = modeget;
        zlen = zlenget;
        ilen = ilenget;
        init();
    }
    void init() {
        Edge.res(size);
        v2n.res(size);
        n2v.res(size);
        Primresult.asn(size, mp(ilen, -1));
    }
    lentype lenplus(lentype l, lentype r) {
        return l + r;
    }
    lentype lenequal(lentype l, lentype r) {
        return l == r;
    }
    lentype lenlcr(lentype l, lentype r) {
        return l < r;
    }
    ll addV(ll vs) {
        size += vs;
        init();
        return size;
    }
    void caddE(ll x, ll y, lentype c) {
        if (mapmode) v2n[x][y] = Edge[x].sz;
        Edge[x].pb(c);
        n2v[x].pb(y);
    }
    void csetE(ll x, ll y, lentype c) {
        if (mapmode) Edge[x][v2n[x][y]] = c;
    }
    void cersE(ll x, ll y) {
        if (mapmode) {
            ll n = v2n[x][y];
            Edge[x][n] = ilen;
            n2v[x][n] = -1;
            v2n[x].ers(y);
        }
    }
    void addE(ll x, ll y, lentype c) {
        caddE(x, y, c);
        if (!mode) caddE(y, x, c);
    }
    void setE(ll x, ll y, lentype c) {
        csetE(x, y, c);
        if (!mode) csetE(y, x, c);
    }
    void ersE(ll x, ll y, lentype c) {
        cersE(x, y, c);
        if (!mode) cersE(y, x, c);
    }
    lentype getE(ll x, ll y) {
        if (mapmode) {
            if (v2n[x].count(y)) {
                return Edge[x][v2n[x][y]];
            }
        }
        return ilen;
    }
    ll getVsz(ll x) {
        return Edge[x].sz;
    }
    pair<lentype, ll> getV(ll x, ll n) {
        if (n >= getVsz(x)) return mp(ilen, -1);
        return mp(Edge[x][n], n2v[x][n]);
    }

    vector<pair<lentype, vll>> Dijk(ll x) {
        vector<pair<lentype, vll>> result(size);
        REP(i, size) {
            result[i].fi = ilen;
            result[i].se = { -1 };
        }
        vll stat(size, 0);
        pair<lentype, ll> now;
        pair<lentype, ll> nowlv;
        SegT<pair<lentype, ll>> Q(size, mp(ilen, -1), [](pair<lentype, ll> l, pair<lentype, ll> r) {
            if (l.se == -1) return r;
            if (r.se == -1) return l;
            return l.fi < r.fi ? l : r;
        });
        Q.set(x, mp(zlen, x));
        result[x].fi = zlen;
        result[x].se = { -1 };
        while (Q.que(0, size - 1).se != -1) {
            now = Q.que(0, size - 1);
            Q.set(now.se, mp(ilen, -1));
            stat[now.se] = 1;
            REP(i, getVsz(now.se)) {
                nowlv = getV(now.se, i);
                if (stat[nowlv.se]) continue;
                if (Q[nowlv.se].se == -1) {
                    result[nowlv.se].fi = lenplus(result[now.se].fi, nowlv.fi);
                    result[nowlv.se].se = { now.se };
                    Q.set(nowlv.se, mp(result[nowlv.se].fi, nowlv.se));
                }
                else {
                    if (lenlcr(lenplus(result[now.se].fi, nowlv.fi), result[nowlv.se].fi)) {
                        result[nowlv.se].fi = lenplus(result[now.se].fi, nowlv.fi);
                        result[nowlv.se].se = { now.se };
                        Q.set(nowlv.se, mp(result[nowlv.se].fi, nowlv.se));
                    }
                    else if (lenequal(lenplus(result[now.se].fi, nowlv.fi), result[nowlv.se].fi)) {
                        result[nowlv.se].se.pb(now.se);
                    }
                }
            }
        }
        return result;
    }

    lentype Prim(ll x = 0) {
        lentype ans = zlen;
        vll stat(size, 0);
        pair<lentype, ll> now;
        pair<lentype, ll> nowlv;
        SegT<pair<lentype, ll>> Q(size, mp(ilen, -1), [](pair<lentype, ll> l, pair<lentype, ll> r) {
            if (l.se == -1) return r;
            if (r.se == -1) return l;
            return l.fi < r.fi ? l : r;
        });
        Q.set(x, mp(zlen, x));
        Primresult[x] = mp(zlen, -1);
        while (Q.que(0, size - 1).se != -1) {
            now = Q.que(0, size - 1);
            Q.set(now.se, mp(ilen, -1));
            stat[now.se] = 1;
            ans = lenplus(ans, Primresult[now.se].fi);
            REP(i, getVsz(now.se)) {
                nowlv = getV(now.se, i);
                if (stat[nowlv.se]) continue;
                if (Q[nowlv.se].se == -1) {
                    Primresult[nowlv.se] = mp(nowlv.fi, now.se);
                    Q.set(nowlv.se, mp(Primresult[nowlv.se].fi, nowlv.se));
                }
                else {
                    if (lenlcr(nowlv.fi, Primresult[nowlv.se].fi)) {
                        Primresult[nowlv.se] = mp(nowlv.fi, now.se);
                        Q.set(nowlv.se, mp(Primresult[nowlv.se].fi, nowlv.se));
                    }
                }
            }
        }
        return ans;
    }

};


/*template <class Type> class DP {

public:

    vector<Type> v;
    Type initv;
    vll size, block;

    DP() {}
    DP(const DP &dp) {}

    template<class... Args> DP(Args... args) {
        block.asn(1, 1);
        Initialize(args...);
    }

    void Initialize(Type initv_) {
        initv = initv_;
        v.asn(block.bk, initv);
    }
    template<class... Args> void Initialize(ll val, Args... args) {
        size.pb(val);
        block.pb(block.bk*val);
        Initialize(args...);
    }



};*/


pl Bezout(ll a, ll b) {
    if (b != 0) {
        pl xy;
        xy = Bezout(b, a%b);
        return mp(xy.se, xy.fi - ((a / b)*xy.se));
    }
    else {
        return mp(1, 0);
    }
}
pl Bez(ll a, ll b, ll c) {
    pl xy;
    ll x, y, z, gc;
    xy = Bezout(a, b);
    gc = gcd(a, b);
    if (c%gc != 0) return mp(-1, -1);
    x = xy.fi*(c / gc); y = xy.se*(c / gc);
    if (x < 0) z = rup(-x, (b / gc));
    if (x >= 0) z = -x / (b / gc);
    x += z * (b / gc);
    y -= z * (a / gc);
    return mp(x, y);
}

ll DigS10(ll n) {
    ll ans = 0;
    while(1) {
        ans += n % 10;
        n /= 10;
        if (!n) break;
    }
    return ans;
}

ll isP(ll n) {
    if (n <= 1) return 0;
    FOR(i, 2, (ll)sqrt(n) + 1) {
        if (n%i == 0) return 0;
    }
    return 1;
}

ll Tot(ll n) {
    if (n <= 0) return 0;
    ll ans = n, x = 2;
    while (x*x <= n) {
        if (n%x == 0) {
            ans -= ans / x;
            while (n%x == 0) n /= x;
        }
        x++;
    }
    if (n > 1) ans -= ans / n;
    return ans;
}

template <class T> struct Sum {
public:
    V<T> v, s;
    ll size;
    Action<T> Add;
    Sum(V<T> v, Action<T> Add = ADD<T>) : v(v), size(v.sz), Add(Add) { Init(); Calc(); }
    void Init() {
        s.asn(size + 1, Add.I);
    }
    void Calc() {
        REP(i, size) s[i + 1] = Add(s[i], v[i]);

    }
    T operator()(ll x) {
        return operator()(0, x);
    }
    T operator()(ll l, ll r) {
        if (l < 0) l = 0;
        if (r >= size) r = size - 1;
        if (l > r) return Add.I;
        return s[r + 1] - s[l];//for ADD
    }
};
using sumll = Sum<ll>;

template <class T> struct Sum2 {
public:
    V<V<T>> v, s;
    ll RowSize, ColumnSize;
    Action<T> Add;
    Sum2(V<V<T>> v, Action<T> Add = ADD<T>) : v(v), RowSize(v.sz), ColumnSize(v.sz ? v[0].sz : 0), Add(Add) { Init(); Calc(); }
    void Init() {
        s.asn(RowSize + 1, V<T>(ColumnSize, Add.I));
    }
    void Calc() {
        REP1(r, RowSize) {
            REP1(c, ColumnSize) {
                s[r][c] = v[r - 1][c - 1] + operator()(r, c - 1) + operator()(r - 1, c) - operator()(r - 1, c - 1);
            }
        }
    }
    T operator()(ll r, ll c) {
        return operator()(0, 0, r, c);
    }
    T operator()(ll r1, ll c1, ll r2, ll c2) {
        if (r1 < 0) r1 = 0;
        if (c1 < 0) c1 = 0;
        if (r2 >= RowSize) r2 = RowSize - 1;
        if (c2 >= ColumnSize) c2 = ColumnSize - 1;
        if (r1 > r2) return Add.I;
        if (c1 > c2) return Add.I;
        return s[r2 + 1][c2 + 1] - s[r2 + 1][c1] - s[r1][c2 + 1] + s[r1][c1];
    }
};
using sum2ll = Sum2<ll>;

template <class T> struct Point2 {
public:
    V<V<T>> v;
    ll h, w;
    Point2() : h(0), w(0) {}
    Point2(ll h, ll w) : h(h), w(w) { asn(h, w); }
    Point2(ll h, ll w, T val) : h(h), w(w) { asn(h, w, val); }
    void assign(ll h, ll w) { v.asn(h, V<T>(w)); }
    void assign(ll h, ll w, ll val) { v.asn(h, V<T>(w, val)); }
    T& operator()(ll h, ll w) { return v[h][w]; }
    T& operator()(pl p) { return v[p.fi][p.se]; }
};
template <class T> using P2 = Point2<T>;

template <ll Mod> struct Modll {
public:
    ll v;
    Modll() : v(0) {}
    Modll(ll _v) { set(_v % Mod + Mod); }
    Modll& set(ll _v) {
        v = (_v < Mod) ? _v : (_v - Mod);
        return *this;
    }
    Modll pow(ll n) const {
        Modll x = *this, ans = 1;
        while (n) {
            if (n & 1) ans *= x;
            x *= x;
            n >>= 1;
        }
        return ans;
    }
    Modll inverse() const { return (*this).pow(Mod - 2); }
    Modll operator+(const Modll& m) const { return Modll().set(v + m.v); }
    Modll operator-(const Modll& m) const { return Modll().set(Mod + v - m.v); }
    Modll operator*(const Modll& m) const { return Modll().set((ull(v) * m.v) % Mod); }
    Modll operator/(const Modll& m) const { return *this * m.inv; }
    Modll& operator+=(const Modll& m) { return *this = *this + m; }
    Modll& operator-=(const Modll& m) { return *this = *this - m; }
    Modll& operator*=(const Modll& m) { return *this = *this * m; }
    Modll& operator/=(const Modll& m) { return *this = *this / m; }
    Modll operator-() const { return Modll(0) - *this; }
    explicit operator bool() const { return v != 0; }
    friend ostream& operator<<(ostream& out, const Modll& m) { return out << m.v; }
};
using mll = Modll<MOD>;
using vmll = V<mll>;
using vvmll = V<vmll>;
using vvvmll = V<vvmll>;

template <class T> T vmin(V<T> v) {
    T tmp = MIN<T>.I;
    ROR(v, i) emin(tmp, i);
    return tmp;
}
template <class T, class... Args> T vmin(V<T> v, Args... args) {
    T tmp = vmin(args...);
    return min(vmin(v), tmp);
}

template <class T> T vmax(V<T> v) {
    T tmp = MAX<T>.I;
    ROR(v, i) emax(tmp, i);
    return tmp;
}
template <class T, class... Args> T vmax(V<T> v, Args... args) {
    T tmp = vmax(args...);
    return max(vmax(v), tmp);
}

template <class T> T vgcd(V<T> v) {
    T tmp = GCD<T>.I;
    ROR(v, i) egcd(tmp, i);
    return tmp;
}
template <class T, class... Args> T vgcd(V<T> v, Args... args) {
    T tmp = vgcd(args...);
    return gcd(vgcd(v), tmp);
}

template <class T> T vlcm(V<T> v) {
    T tmp = LCM<T>.I;
    ROR(v, i) elcm(tmp, i);
    return tmp;
}
template <class T, class... Args> T vlcm(V<T> v, Args... args) {
    T tmp = vlcm(args...);
    return lcm(vlcm(v), tmp);
}

vmll MFactMemo(2, 1);
vmll MIFactMemo(2, 1);

mll mFact(ll n) {
    if (MFactMemo.sz <= n) {
        ll oldsize = MFactMemo.sz;
        MFactMemo.res(n + 1, 1);
        FOR(i, oldsize, n) MFactMemo[i] = MFactMemo[i - 1] * i;
    }
    return MFactMemo[n];
}
mll miFact(ll n) {
    if (MIFactMemo.sz <= n) {
        ll oldsize = MIFactMemo.sz;
        MIFactMemo.res(n + 1, 1);
        MIFactMemo.bk = mFact(n).inv;
        rFOR(i, oldsize + 1, n) MIFactMemo[i - 1] = MIFactMemo[i] * i;
    }
    return MIFactMemo[n];
}
mll mComb(ll n, ll k) {
    if (n < 0 || k < 0 || n < k) return 0;
    return mFact(n) * miFact(k) * miFact(n - k);
}

ll LIS(vll v, ll m = 0) {

    if (v.sz > 0) {
        ll ans = 0;
        vll dp(v.sz, INF);
        FOR(i, 0, v.sz - 1) {
            dp[m ? UPB(dp, v[i]) : LOWB(dp, v[i])] = v[i];
        }
        FOR(i, 0, v.sz - 1) {
            if (dp[i] == INF) break;
            ans++;
        }
        return ans;
    }
    else {
        return 0;
    }

}

void PCmprs(vll& v) {
    if (v.sz == 0) return;
    vll vv(v);
    IOTA(vv, 0);
    sort(vv.bgn, vv.en, [&](ll v1, ll v2) {return v[v1] < v[v2]; });
    IOTA(v, 0);
    sort(v.bgn, v.en, [&](ll v1, ll v2) {return vv[v1] < vv[v2]; });
    return;
}

ll BblCnt(vll v) {
    PCmprs(v);
    SegT<ll> b(v.sz, 0);
    ll ans = 0;
    REP(i, v.sz) {
        ans += (i - b.que(0, v[i]));
        b.add(v[i], 1);
    }
    return ans;
}

ll Bsrch(function<bool(ll x)> f, ll mi, ll ma) {
    ll ng = mi - 1, ok = ma, mid;
    while (ok - ng > 1) {
        mid = (ng + ok) / 2;
        (f(mid) ? ok : ng) = mid;
    }
    return ok;
}

template<class T> V<V<T>> MultiMatrix(V<V<T>> A, V<V<T>> B, Action<T> Mul = MUL<T>, Action<T> Add = ADD<T>) {
    V<V<T>> ans;
    ll ii = A.sz;
    if (!ii) return ans;
    ll jj = A[0].sz;
    if (!jj) return ans;
    ll jj2 = B.sz;
    if (!jj2) return ans;
    if (jj != jj2) return ans;
    ll kk = B[0].sz;
    if (!kk) return ans;
    ans.asn(ii, V<T>(kk, 0));
    REP(i, ii) {
        REP(k, kk) {
            REP(j, jj) {
                ans[i][k] = Add(ans[i][k], Mul(A[i][j], B[j][k]));
            }
        }
    }
    return ans;
}

vvll CombMemo(1000, vll(1000, -1));

ll Comb(ll n, ll k) {
    if ((n < 0) || (k < 0)) return 0;
    if (CombMemo[n][k] == -1) {
        if (n < k) CombMemo[n][k] = 0;
        else {
            if (n == 0) CombMemo[n][k] = 1;
            else if (k == 0) CombMemo[n][k] = 1;
            else if (n == k) CombMemo[n][k] = 1;
            else CombMemo[n][k] = Comb(n - 1, k - 1) + Comb(n - 1, k);
        }
    }
    return CombMemo[n][k];
}

template<class T> map<T, ll> Dict(V<T> v) {
    map<T, ll> m;
    if (!v.sz) return m;
    SORT(v);
    UNIQUE(v);
    REP(i, v.sz) {
        m[v[i]] = i;
    }
    return m;
}

template<class T> vll Cmprs(V<T> v) {
    auto m = Dict(v);
    vll ans(v.sz);
    REP(i,v.sz) {
        ans[i] = m[v[i]];
    }
    return ans;
}

template <class T> auto vecn(T val) { return val; }
template <class... Args> auto vecn(ll val, Args... args) {
    auto tmp = vecn(args...);
    return V<decltype(tmp)>(val, tmp);
}

void Solve();

signed main() { 
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << setprecision(20) << fixed;
    Solve();
}

void Solve() {
  li(n,k);
  double r=0,d;
  ll b;
  REP1(i,n){
    d=(double)1;
    b=i;
    while(b<k){
      d/=(double)2;
      b*=2;
    }
    r+=d/(double)n;
  }
  cout << r;
}

















Measured memory_bytes: 8064