AtCoder Express

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

inputFormat

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

outputFormat

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

样例

1
9
10

20.250000000000000000