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 PYTHON):

Solution:
#!/usr/bin/python
# coding: utf-8

# AtCoder Beginner Contest 076
# D - AtCoder Express

from sys import stdin

# 移動距離を返す
def solve(ts, vs):
  if len(ts) == 0: return 0.0

  time  = sum(ts)  # 総時間
  limit = min(vs)  # 制限速度の下限
  if time / 2.0 <= limit:     # limitまで加速する時間がない
    return (time / 2.0) ** 2  # 半分の時間で加速，残りで減速

  # 制限速度まで達した区間がある
  if limit == 0:
    # その区間を除いて計算
    vzero = [i for i in range(len(vs)) if vs[i] == 0][0]
    return solve(ts[:vzero],     vs[:vzero]) + \
           solve(ts[vzero + 1:], vs[vzero + 1:])

  # 最初に1加速, 最後に1減速
  ts[0]  -= 1
  ts[-1] -= 1
  # 長さ0の区間を削除
  if ts[-1] == 0:
    ts = ts[:-1]
    vs = vs[:-1]
  if ts[0] == 0:
    ts = ts[1:]
    vs = vs[1:]
  # 制限時間を1減らす (= 現在速度を1増やす)
  vs = [v - 1 for v in vs]
  # 1加速, 速度1でtime-2秒移動, 1減速, の移動距離
  dist = time - 2 + 1
  return dist + solve(ts, vs)

# 文字列をintのリストに分解する
def splitToInt(line):
  words = line.strip().split()
  return [int(w) for w in words]

N = int(stdin.readline())
ts = splitToInt(stdin.readline())
vs = splitToInt(stdin.readline())
if len(ts) != N or len(vs) != N: raise
print(solve(ts, vs))