Problem:
Score : 1300 points

Problem Statement

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}.

Sample Input 1

5
4 2 6 3 5
1 1 1 1 1

Sample Output 1

4.700000000000

Sample Input 2

4
100 0 100 0
0 100 0 100

Sample Output 2

50.000000000000

Sample Input 3

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

Sample Output 3

7047.142857142857

Sample Input 4

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

Sample Output 4

815899161079.400024414062

Solution in Rust (1.42.0) (judge verdict: Time Limit Exceeded)

#[allow(unused_imports)]
use std::{
    i64,
    f64,
    cmp,
};

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };

    (stream = $s:expr, $($r:tt)*) => {
        input_inner!{$s, $($r)*}
    };

    ($($r:tt)*) => {
        use std::io::*;
        let mut stream = stdin();
        let stream = &mut stream;
        input_inner!{stream, $($r)*}
    };
}

macro_rules! input_inner {
    ($stream:expr) => {};
    ($stream:expr, ) => {};

    ($stream:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($stream, $t);
        input_inner!{$stream $($r)*}
    };
}

macro_rules! read_value {
    ($stream:expr, ( $($t:tt),* )) => {
        ( $(read_value!($stream, $t)),* )
    };

    ($stream:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($stream, $t)).collect::<Vec<_>>()
    };

    ($stream:expr, chars) => {
        read_value!($stream, String).chars().collect::<Vec<char>>()
    };

    ($stream:expr, usize1) => {
        read_value!($stream, usize) - 1
    };

    ($stream:expr, $t:ty) => {
        $stream.by_ref().bytes()
            .map(|c| c.unwrap() as char)
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>()
            .parse::<$t>()
            .expect("Parse error")
    };
}

fn main() {
    input!{n: usize};
    input!{
        a: [i64; n],
        b: [f64; n]
    }
    let mut prev_g: Vec<f64> = vec![*a.iter().max().unwrap() as f64; n];
    let a = a.iter().map(|a| {*a as f64}).collect::<Vec<f64>>();
    let mut new_g: Vec<f64> = vec![0.0; n];
    let mut prev_sum: f64 = prev_g.iter().sum();
    let mut new_sum: f64;

    loop {
        for i in 0 .. n {
            let next_e = 0.5 * (prev_g[(n + i - 1) % n] + prev_g[(i + 1) % n]) - b[i];
            new_g[i] = if a[i] >= next_e {
                a[i]
            } else {
                next_e
            };

            if (next_e - a[i]).abs() < 1e-2 {
                new_g[i] = a[i];
            }
        }

        new_sum = new_g.iter().sum();
        if (new_sum - prev_sum).abs() <= prev_sum.abs() * 1e-12 {
            println!("{}", new_sum / (n as f64));
            break;
        }

        prev_g = new_g.clone();
        prev_sum = new_sum;
    }
}