Problem: In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.

In the plan developed by the president Takahashi, the trains will run as follows:

* A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.
* In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.



According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.

Find the maximum possible distance that a train can cover in the run.

Constraints

* 1 \leq N \leq 100
* 1 \leq t_i \leq 200
* 1 \leq v_i \leq 100
* All input values are integers.

Input

Input is given from Standard Input in the following format:


N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N


Output

Print the maximum possible that a train can cover in the run.
Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

Examples

Input

1
100
30


Output

2100.000000000000000


Input

2
60 50
34 38


Output

2632.000000000000000


Input

3
12 14 2
6 2 7


Output

76.000000000000000


Input

1
9
10


Output

20.250000000000000000


Input

10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70


Output

20291.000000000000

Solution (in CPP):

Solution:
#include "bits/stdc++.h"
#define in std::cin
#define out std::cout
#define rep(i,N) for(int i=0;i<N;++i)
typedef long long int LL;
 
const LL inf = 11234567;
LL N;
LL t[112], v[112];
LL seek, last, lim[41234], memo[41234][234];
 
LL dp(LL ti, LL sp)
{
	if (sp < 0) { return -inf; }
	if (ti == 0) {
		if(sp == 0)return 0LL; 
		return -inf;
	}
	if (sp > lim[ti - 1])return -inf;
	if (memo[ti][sp] != -1) { return memo[ti][sp]; }
	LL res = dp(ti - 1, sp) + sp + sp;
	{ res = std::max(res, dp(ti - 1, sp - 1) + sp + (sp - 1)); }
	if(sp+1<=lim[ti-1]){ res = std::max(res, dp(ti - 1, sp + 1) + sp + (sp + 1)); }
	return memo[ti][sp] = res;
}
 
int main()
{
	in >> N;
	rep(i, N) { in >> t[i]; }
	rep(i, N) { in >> v[i]; }
 
	memset(memo, -1, sizeof(memo));
	rep(i, N)
	{
		t[i] *= 2;
		v[i] *= 2;
		rep(j, t[i]) { lim[seek++] = v[i]; }
		last += t[i];
	}
 
	out << std::fixed << std::setprecision(15);
	out << (double)dp(last, 0) / (double)8 << std::endl;
	return 0;
}