Problem: You are playing a game and your goal is to maximize your expected gain. At the beginning of the game, a pawn is put, uniformly at random, at a position p\in\\{1,2,\dots, N\\}. The N positions are arranged on a circle (so that 1 is between N and 2).

The game consists of turns. At each turn you can either end the game, and get A_p dollars (where p is the current position of the pawn), or pay B_p dollar to keep playing. If you decide to keep playing, the pawn is randomly moved to one of the two adjacent positions p-1, p+1 (with the identifications 0 = N and N+1=1).

What is the expected gain of an optimal strategy?

Note: The "expected gain of an optimal strategy" shall be defined as the supremum of the expected gain among all strategies such that the game ends in a finite number of turns.

Constraints

* 2 \le N \le 200,000
* 0 \le A_p \le 10^{12} for any p = 1,\ldots, N
* 0 \le B_p \le 100 for any p = 1, \ldots, N
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N
A_1 A_2 \cdots A_N
B_1 B_2 \cdots B_N


Output

Print a single real number, the expected gain of an optimal strategy. Your answer will be considered correct if its relative or absolute error does not exceed 10^{-10}.

Examples

Input

5
4 2 6 3 5
1 1 1 1 1


Output

4.700000000000


Input

4
100 0 100 0
0 100 0 100


Output

50.000000000000


Input

14
4839 5400 6231 5800 6001 5200 6350 7133 7986 8012 7537 7013 6477 5912
34 54 61 32 52 61 21 43 65 12 45 21 1 4


Output

7047.142857142857


Input

10
470606482521 533212137322 116718867454 746976621474 457112271419 815899162072 641324977314 88281100571 9231169966 455007126951
26 83 30 59 100 88 84 91 54 61


Output

815899161079.400024414062

Solution (in CPP):

Solution:
/**
 * code generated by JHelper
 * More info: https://github.com/AlexeyDmitriev/JHelper
 * @author Egor Kulikov
 */





#include <bits/stdc++.h>

using namespace std;

#define all(v) (v).begin(), (v).end()

using ll = long long;
using ull = unsigned long long;
using li = __int128;
using uli = unsigned __int128;
using ld = long double;
using pii = pair<int, int>;
using vi = vector<int>;

void doReplace() {
}

template <typename T, typename U, typename...Vs>
void doReplace(T& t, const U& u, Vs&& ...vs) {
    t = u;
    doReplace(vs...);
}

template <typename T, typename...Us>
T minim(T& was, const T& cand, Us&& ...us) {
    if (was > cand) {
        was = cand;
        doReplace(us...);
    }
    return was;
}

template <typename T, typename...Us>
T maxim(T& was, const T& cand, Us&& ...us) {
    if (was < cand) {
        was = cand;
        doReplace(us...);
    }
    return was;
}


template <typename D>
D dPower(D base, ll exponent) {
    if (exponent < 0) {
        return dPower(1 / base, -exponent);
    }
    if (exponent == 0) {
        return 1;
    }
    if ((exponent & 1) == 1) {
        return dPower(base, exponent - 1) * base;
    } else {
        D res = dPower(base, exponent >> 1);
        return res * res;
    }
}


class NumberIterator : iterator<forward_iterator_tag, int> {
public:
    int v;

    NumberIterator(int v) : v(v) {}

    operator int&() { return v; }

    int operator*() { return v; }
};

class range : pii {
public:
    range(int begin, int end) : pii(begin, max(begin, end)) {}

    range(int n) : pii(0, max(0, n)) {}

    NumberIterator begin() {
        return first;
    }

    NumberIterator end() {
        return second;
    }
};


template <typename T>
class arr {
    T* b;
    int n;

    void allocate(int sz) {
#ifdef LOCAL
        if (sz < 0) {
            throw "bad alloc";
        }
#endif
        n = sz;
        if (sz > 0) {
            b = (T*) (::operator new(sz * sizeof(T)));
        } else {
            b = nullptr;
        }
    }

public:
    arr(int n = 0) {
        allocate(n);
#ifdef LOCAL
        view();
#endif
    }

    arr(int n, const T& init) {
        allocate(n);
        for (int i : range(n)) {
            ::new((void*) (b + i)) T(init);
        }
#ifdef LOCAL
        view();
#endif
    }

    arr(initializer_list<T> l) : arr(l.size()) {
        if (n > 0) {
            memcpy(b, l.begin(), n * sizeof(T));
        }
    }

    arr(T* b, int n) : arr(b, b + n) {}

    arr(T* b, T* e) : b(b), n(e - b) {}

    int size() const {
        return n;
    }

    T* begin() {
        return b;
    }

    const T* begin() const {
        return b;
    }

    T* end() {
        return b + n;
    }

    const T* end() const {
        return b + n;
    }

    arr<T> clone() const {
        arr<T> res(n);
        copy(b, b + n, res.begin());
        return res;
    }

    T& operator[](int at) {
#ifdef LOCAL
        if (at < 0 || at >= n) {
            throw "Out of bounds";
        }
#endif
        return b[at];
    }

    const T& operator[](int at) const {
#ifdef LOCAL
        if (at < 0 || at >= n) {
            throw "Out of bounds";
        }
#endif
        return b[at];
    }

    bool operator==(const arr& other) const {
        if (n != other.n) {
            return false;
        }
        for (int i : range(n)) {
            if (b[i] != other.b[i]) {
                return false;
            }
        }
        return true;
    }

    vector<T> view() {
        return vector<T>(b, b + min(n, 50));
    }
};

typedef arr<int> arri;

void decreaseByOne() {}

template <typename T, class...Vs>
void decreaseByOne(arr<T>& array, Vs& ...vs) {
    int n = array.size();
    for (int i : range(n)) {
        array[i]--;
    }
    decreaseByOne(vs...);
}

template <typename T, typename U>
void decreaseByOne(arr<pair<T, U>>& v) {
    for (auto& p : v) {
        p.first--;
        p.second--;
    }
}


template <typename T>
class arr2d {
    T* b;
    int d1;
    int d2;
    int sz;

    void allocate(int n) {
#ifdef LOCAL
        if (n < 0) {
            throw "bad alloc";
        }
#endif
        if (n > 0) {
            b = (T*) (::operator new(n * sizeof(T)));
        } else {
            b = nullptr;
        }
    }

public:
    arr2d(int d1 = 0, int d2 = 0) : d1(d1), d2(d2), sz(d1 * d2) {
#ifdef LOCAL
        if (d1 < 0 || d2 < 0) {
            throw "bad alloc";
        }
#endif
        allocate(sz);
#ifdef LOCAL
        view();
#endif
    }

    arr2d(int d1, int d2, const T& init) : d1(d1), d2(d2), sz(d1 * d2) {
#ifdef LOCAL
        if (d1 < 0 || d2 < 0) {
            throw "bad alloc";
        }
#endif
        allocate(sz);
        for (int i : range(sz)) {
            ::new((void*) (b + i)) T(init);
        }
#ifdef LOCAL
        view();
#endif
    }

    arr2d(T* b, int d1, int d2) : b(b), d1(d1), d2(d2), sz(d1 * d2) {}

    int size() const {
        return sz;
    }

    int dim1() const {
        return d1;
    }

    int dim2() const {
        return d2;
    }

    T* begin() {
        return b;
    }

    T* end() {
        return b + sz;
    }

    T& operator()(int i1, int i2) {
#ifdef LOCAL
        if (i1 < 0 || i1 >= d1 || i2 < 0 || i2 >= d2) {
            throw "Out of bounds";
        }
#endif
        return b[i1 * d2 + i2];
    }

    const T& operator()(int i1, int i2) const {
#ifdef LOCAL
        if (i1 < 0 || i1 >= d1 || i2 < 0 || i2 >= d2) {
            throw "Out of bounds";
        }
#endif
        return b[i1 * d2 + i2];
    }

    T& operator[](const pii& p) {
        return operator()(p.first, p.second);
    }

    const T& operator[](const pii& p) const {
        return operator()(p.first, p.second);
    }

    arr<T> operator[](int at) {
#ifdef LOCAL
        if (at < 0 || at >= d1) {
            throw "Out of bounds";
        }
#endif
        return arr<T>(b + at * d2, d2);
    }

    vector<vector<T>> view() {
        vector<vector<T>> res(min(d1, 50));
        for (int i : range(res.size())) {
            res[i] = (*this)[i].view();
        }
        return res;
    }

    arr2d<T> clone() {
        arr2d<T> res(d1, d2);
        copy(b, b + sz, res.b);
        return res;
    }
};


inline bool isWhitespace(int c) {
    return isspace(c) || c == EOF;
}

class Input {
private:
    bool exhausted = false;
    int bufSize = 4096;
    char* buf = new char[bufSize];
    int bufRead = 0;
    int bufAt = 0;

public:
    inline int get() {
        if (exhausted) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return EOF;
        }
        if (bufRead == bufAt) {
            bufRead = fread(buf, sizeof(char), bufSize, stdin);
            bufAt = 0;
        }
        if (bufRead == bufAt) {
            exhausted = true;
            return EOF;
        }
        return buf[bufAt++];
    }

private:
    template <typename T>
    inline T readInteger() {
        int c = skipWhitespace();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = get();
        }
        T res = 0;
        do {
            if (!isdigit(c)) {
#ifdef LOCAL
                throw "Number format error";
#endif
                return sgn * res;
            }
            res *= 10;
            res += c - '0';
            c = get();
        } while (!isWhitespace(c));
        return res * sgn;
    }

    void initArrays(int) {}

    template <typename T, class...Vs>
    void initArrays(int n, arr<T>& array, Vs& ...vs) {
        array = arr<T>(n, T());
        initArrays(n, vs...);
    }

    template <typename T, class...Vs>
    void initArrays(int n, vector<T>&, Vs& ...vs) {
        initArrays(n, vs...);
    }

    void readImpl(int) {}

    template <typename T, class...Vs>
    void readImpl(int i, arr<T>& arr, Vs& ...vs) {
        arr[i] = readType<T>();
        readImpl(i, vs...);
    }

    template <typename T, class...Vs>
    void readImpl(int i, vector<T>& arr, Vs& ...vs) {
        arr.push_back(readType<T>());
        readImpl(i, vs...);
    }

public:
    inline int skipWhitespace() {
        int c;
        while (isWhitespace(c = get()) && c != EOF);
        return c;
    }

    inline int readInt() {
        return readInteger<int>();
    }

    inline ll readLong() {
        return readInteger<ll>();
    }

    string readString() {
        int c = skipWhitespace();
        if (c == EOF) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return "";
        }
        string res;
        do {
            res.push_back(c);
        } while (!isWhitespace(c = get()));
        return res;
    }

    arri readIntArray(int size) {
        return readArray<int>(size);
    }

    arr<ll> readLongArray(int size) {
        return readArray<ll>(size);
    }

    arr<double> readDoubleArray(int size) {
        return readArray<double>(size);
    }

    arr<string> readStringArray(int size) {
        return readArray<string>(size);
    }

    arr<char> readCharArray(int size) {
        return readArray<char>(size);
    }

    template <typename T>
    T readType() {
        throw "Operation not supported";
    }

    template <typename U, typename V>
    pair<U, V> readType() {
        U first = readType<U>();
        V second = readType<V>();
        return make_pair(first, second);
    }

    template <typename T>
    arr<T> readArray(int n) {
        arr<T> res(n, T());
        for (int i : range(n)) {
            res[i] = readType<T>();
        }
        return res;
    }


    template <class...Vs>
    void readArrays(int n, Vs& ...vs) {
        initArrays(n, vs...);
        for (int i : range(n)) {
            readImpl(i, vs...);
        }
    }

    template <typename U, typename V>
    arr<pair<U, V>> readArray(int n) {
        arr<pair<U, V>> res(n);
        for (int i : range(n)) {
            res[i] = readType<U, V>();
        }
        return res;
    }

    template <typename T>
    arr2d<T> readTable(int rows, int cols) {
        arr2d<T> result(rows, cols);
        for (int i : range(rows)) {
            for (int j : range(cols)) {
                result(i, j) = readType<T>();
            }
        }
        return result;
    }

    arr2d<int> readIntTable(int rows, int cols) {
        return readTable<int>(rows, cols);
    }

    arr2d<char> readCharTable(int rows, int cols) {
        return readTable<char>(rows, cols);
    }

    string readLine() {
        int c = get();
        if (c == EOF) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return "";
        }
        string res;
        do {
            res.push_back(c);
            c = get();
        } while (c != '\n' && c != '\r' && c != EOF);
        return res;
    }

    inline double readDouble() {
        int c = skipWhitespace();
        int sgn = 1;
        if (c == '-') {
            sgn = -1;
            c = get();
        }
        double res = 0;
        do {
            if (tolower(c) == 'e') {
                return sgn * res * dPower(double(10), readInt());
            }
            if (!isdigit(c)) {
#ifdef LOCAL
                throw "Number format error";
#endif
                return sgn * res;
            }
            res *= 10;
            res += c - '0';
            c = get();
        } while (!isWhitespace(c) && c != '.');
        if (c == '.') {
            double add = 0;
            int length = 0;
            c = get();
            do {
                if (tolower(c) == 'e') {
                    return sgn * (res + add * dPower(double(10), -length)) * dPower(double(10), readInt());
                }
                if (!isdigit(c)) {
#ifdef LOCAL
                    throw "Number format error";
#endif
                    res += add * dPower(10, -length);
                    return res * sgn;
                }
                add *= 10;
                add += c - '0';
                length++;
                c = get();
            } while (!isWhitespace(c));
            res += add * dPower(double(10), -length);
        }
        return res * sgn;
    }

    inline char readChar() {
        int c = skipWhitespace();
        if (c == EOF) {
#ifdef LOCAL
            throw "Input exhausted";
#endif
            return 0;
        }
        return c;
    }

    inline bool isExhausted() { return exhausted; }

    inline void setBufSize(int newBufSize) {
        if (newBufSize > bufSize) {
            char* newBuf = new char[newBufSize];
            memcpy(newBuf, buf, bufSize);
            buf = newBuf;
        }
        bufSize = newBufSize;
    }
};

template <>
inline double Input::readType() {
    return readDouble();
}

template <>
inline int Input::readType() {
    return readInt();
}

template <>
inline ll Input::readType() {
    return readLong();
}

template <>
inline char Input::readType() {
    return readChar();
}

template <>
inline string Input::readType() {
    return readString();
}

Input in;


class Output {
private:
    ostream* out;

    template <typename T>
    inline void printSingle(const T& value) {
        *out << value;
    }

    template <typename T>
    void printSingle(const vector<T>& array) {
        size_t n = array.size();
        for (int i : range(n)) {
            *out << array[i];
            if (i + 1 != n) {
                *out << ' ';
            }
        }
    }

    template <typename T>
    void printSingle(const arr<T>& array) {
        int n = array.size();
        for (int i : range(n)) {
            *out << array[i];
            if (i + 1 != n) {
                *out << ' ';
            }
        }
    }

    template <typename T>
    void printSingle(const arr2d<T>& array) {
        size_t n = array.dim1();
        size_t m = array.dim2();
        for (int i : range(n)) {
            for (int j : range(m)) {
                *out << array(i, j);
                if (j + 1 != m) {
                    *out << ' ';
                }
            }
            if (i + 1 != n) {
                *out << '\n';
            }
        }
    }

    template <typename T, typename U>
    inline void printSingle(const pair<T, U>& value) {
        out << value.first << ' ' << value.second;
    }

public:
    bool autoflush;

    Output(ostream& out, bool autoflush) : out(&out), autoflush(autoflush) {
        out << fixed << setprecision(20);
    }

    void setOut(ostream& nOut) {
        out = &nOut;
    }

    inline void print() {}

    template <typename T, typename...Targs>
    inline void print(const T& first, const Targs... args) {
        printSingle(first);
        if (sizeof...(args) != 0) {
            *out << ' ';
            print(args...);
        }
        if (autoflush) {
            flush();
        }
    }

    template <typename...Targs>
    inline void printLine(const Targs... args) {
        print(args...);
        *out << '\n';
        if (autoflush) {
            flush();
        }
    }

    inline void flush() {
        out->flush();
    }

    inline void setPrecision(int digs) {
        *out << fixed << setprecision(digs);
    }
};

Output out(cout, false);
Output err(cerr, true);


template <typename T>
class FenwickTree {
    arr<T> value;

    T get(int to) const {
        minim(to, int(value.size()) - 1);
        T result = 0;
        while (to >= 0) {
            result += value[to];
            to = (to & (to + 1)) - 1;
        }
        return result;
    }

public:
    FenwickTree(int size) {
        value = arr<T>(size, 0);
    }

    void add(int at, T val) {
        while (at < value.size()) {
            value[at] += val;
            at = at | (at + 1);
        }
    }

    T get(int from, int to) const {
        if (from >= to) {
            return 0;
        }
        return get(to - 1) - get(from - 1);
    }

    void clear() {
        fill(all(value), 0);
    }
};


template <typename T>
class que : public queue<T> {
    using parent = queue<T>;
public:
    que() : parent() {}

    que(const que<T>& q) : parent(q) {}

    que(que<T>&& q) noexcept: parent(move(q)) {}

    T pop() {
#ifdef LOCAL
        if (parent::empty()) {
            throw "Pop on empty queue";
        }
#endif
        T res = parent::front();
        parent::pop();
        return res;
    }

    que<T>& operator=(que<T>&& __x) noexcept {
        parent::operator=(__x);
        return *this;
    }

    que<T>& operator=(const que<T>& __x) {
        parent::operator=(__x);
        return *this;
    }
};

using qi = que<int>;


template <typename T>
inline void unique(vector<T>& v) {
    v.resize(unique(all(v)) - v.begin());
}

arri createOrder(int n) {
    arri order(n);
    for (int i = 0; i < n; i++) {
        order[i] = i;
    }
    return order;
}

arri inverse(const arri& p) {
    arri res(p.size());
    for (int i : range(p.size())) {
        res[p[i]] = i;
    }
    return res;
}

template <typename T, typename Iterator>
inline void addAll(vector<T>& v, Iterator begin, Iterator end) {
    v.insert(v.end(), begin, end);
}

template <class Collection, typename Iterator>
inline void addAll(Collection& v, Iterator begin, Iterator end) {
    v.insert(begin, end);
}

template <typename Iterator>
arri getQty(Iterator begin, Iterator end, int length) {
    arri res(length, 0);
    for (Iterator it = begin; it != end; it++) {
        res[*it]++;
    }
    return res;
}

template <typename Iterator>
arri getQty(Iterator begin, Iterator end) {
    return getQty(begin, end, *max_element(begin, end) + 1);
}

template <class Collection>
void collect(Collection&) {}

template <class Collection, class Other, class ...Vs>
void collect(Collection& all, Other& a, Vs& ...vs) {
    addAll(all, all(a));
    collect(all, vs...);
}

void replace(const vi&) {}

template <class ...Vs>
void replace(const vi& all, vi& a, Vs& ...vs) {
    for (int& i : a) {
        i = lower_bound(all(all), i) - all.begin();
    }
    replace(all, vs...);
}

template <class ...Vs>
void replace(const vi& all, arri& a, Vs& ...vs) {
    for (int& i : a) {
        i = lower_bound(all(all), i) - all.begin();
    }
    replace(all, vs...);
}

template <class ...Vs>
vi compress(Vs& ...vs) {
    vi vals;
    collect(vals, vs...);
    sort(all(vals));
    unique(vals);
    replace(vals, vs...);
    return vals;
}


class ERandomPawn {
public:
    void solve() {
        int n = in.readInt();
        auto a = in.readLongArray(n);
        auto b = in.readIntArray(n);

        int at = max_element(all(a)) - a.begin();
        arr<ll> aa(n + 1);
        arri bb(n + 1);
        for (int i : range(n - at)) {
            aa[i] = a[at + i];
            bb[i] = b[at + i];
        }
        for (int i : range(at)) {
            aa[i + n - at] = a[i];
            bb[i + n - at] = b[i];
        }
        aa[n] = a[at];
        bb[n] = b[at];
        a = aa;
        b = bb;
        FenwickTree<ll> sumB(n);
        FenwickTree<ll> sumBI(n);
        for (int i : range(n)) {
            sumB.add(i, b[i]);
            sumBI.add(i, i * b[i]);
        }
        auto f = [&](int i, int left, int right) -> ll {
            int k = right - left;
            int j = i - left;
            ll base = j * a[right] + (k - j) * a[left];
            ll lb = 2 * ll(j) * k * sumB.get(i, right) -
                    2 * j * (sumBI.get(i, right) - left * sumB.get(i, right));
            ll rb = 2 * (k - j) * (sumBI.get(left + 1, i) - left * sumB.get(left + 1, i));
            ll c = base - (lb + rb);
            return c;
        };
        /*set<int> poi;
        for (int i : range(n + 1)) {
            poi.insert(i);
        }
        que<int> toCheck;
        arr<bool> inQueue(n + 1, true);
        for (int i : range(1, n)) {
            toCheck.push(i);
        }
        while (!toCheck.empty()) {
            int i = toCheck.pop();
            inQueue[i] = false;
            poi.erase(i);
            auto it = poi.lower_bound(i);
            int right = *it;
            it--;
            int left = *it;
            ld c = f(i, left, right);
            if (c >= a[i]) {
                if (!inQueue[left]) {
                    inQueue[left] = true;
                    toCheck.push(left);
                }
                if (!inQueue[right]) {
                    inQueue[right] = true;
                    toCheck.push(right);
                }
            } else {
                poi.insert(i);
            }
        }*/

/*        poi.insert(0);
        poi.insert(n);
        arri order(n - 1);
        for (int i : range(1, n)) {
            order[i - 1] = i;
        }
        sort(all(order), [&](int x, int y) -> bool {
            return a[x] > a[y];
        });
//        ld answer = a[0];
        for (int i : order) {
            auto it = poi.lower_bound(i);
            int right = *it;
            it--;
            int left = *it;
            int k = right - left;
            int j = i - left;
            ld base = ld(j) / k * a[right] + ld(k - j) / k * a[left];
            ld lb = 2 * j * k * sumB.get(i, right) -
                        2 * j * (sumBI.get(i, right) - left * sumB.get(i, right));
            ld rb = 2 * (k - j) * (sumBI.get(left + 1, i) - left * sumB.get(left + 1, i));
            ld c = base - (lb + rb) / k;
            if (c < a[i]) {
                poi.insert(i);
//                answer += a[i];
            } else {
//                answer += c;
            }
        }*/
        vi q;
        for (int i : range(n + 1)) {
            while (q.size() >= 2) {
                int last = q.back();
                int before = q[q.size() - 2];
                if (f(last, before, i) > a[last] * (i - before)) {
                    q.pop_back();
                } else {
                    break;
                }
            }
            q.push_back(i);
        }
        set<int> poi;
        addAll(poi, all(q));
        ld answer = 0;
        for (int i : range(n)) {
            if (poi.count(i)) {
                answer += a[i];
            } else {
                auto it = poi.lower_bound(i);
                int right = *it;
                it--;
                int left = *it;
//                int k = right - left;
//                int j = i - left;
//                ld base = ld(j) / k * a[right] + ld(k - j) / k * a[left];
//                ld lb = 2 * j * k * sumB.get(i, right) -
//                        2 * j * (sumBI.get(i, right) - left * sumB.get(i, right));
//                ld rb = 2 * (k - j) * (sumBI.get(left + 1, i) - left * sumB.get(left + 1, i));
//                ld c = base - (lb + rb) / k;
                answer += ld(f(i, left, right)) / (right - left);
            }
        }
        answer /= n;
        out.setPrecision(20);
        out.printLine(answer);
    }
};


int main() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
#ifdef LOCAL_RELEASE
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
    auto time = clock();
#endif
    ERandomPawn solver;


    solver.solve();
    fflush(stdout);
#ifdef LOCAL_RELEASE
    cerr << clock() - time << endl;
#endif
    return 0;
}