Maximum Sum of Products

Description:

Input:

The first line contains one integer $$$n$$$ ($$$1 \le n \le 5000$$$).

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^7$$$).

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 10^7$$$).

Output:

Print single integer — maximum possible sum after reversing at most one subarray (continuous subsegment) of $$$a$$$.

Sample Input:

5
2 3 2 1 3
1 3 2 4 2

Sample Output:

29

Sample Input:

2
13 37
2 4

Sample Output:

174

Sample Input:

6
1 8 7 6 3 6
5 9 6 8 8 6

Sample Output:

235

Note:

In the first example, you can reverse the subarray $$$[4, 5]$$$. Then $$$a = [2, 3, 2, 3, 1]$$$ and $$$2 \cdot 1 + 3 \cdot 3 + 2 \cdot 2 + 3 \cdot 4 + 1 \cdot 2 = 29$$$.

In the second example, you don't need to use the reverse operation. $$$13 \cdot 2 + 37 \cdot 4 = 174$$$.

In the third example, you can reverse the subarray $$$[3, 5]$$$. Then $$$a = [1, 8, 3, 6, 7, 6]$$$ and $$$1 \cdot 5 + 8 \cdot 9 + 3 \cdot 6 + 6 \cdot 8 + 7 \cdot 8 + 6 \cdot 6 = 235$$$.