Problem Statement

We have K sequences.
The i-th sequence A_i has the length of N_i.
Takahashi and Aoki will play a game using these.
They will alternately do the following operation until the length of every sequence becomes 1:

• Choose a sequence of length at least 2, and delete its first or last element.

Takahashi goes first. Takahashi wants to maximize the sum of the K elements remaining in the end, while Aoki wants to minimize it.
Find the sum of the K elements remaining in the end when both players play optimally.

Input

Input is given from Standard Input in the following format:
K
$N_1 A_{1, 1} A_{1, 2} \cdots A_{1, N_1}$
$N_2 A_{2, 1} A_{2, 2} \cdots A_{2, N_2}$
$\vdots$
$N_K A_{K, 1} A_{K, 2} \cdots A_{K, N_K}$

Output

Print the answer.

Constraints

• All values in input are integers.
• $1 \leq K \leq 2 \times 10^5$
• $1 \leq N_i$
• $\sum_i N_i \leq 2 \times 10^5$
• $1 \leq A_{i, j} \leq 10^9$

Sample Input 1

2
3 1 2 3
2 1 10

Sample Output 1

12

Here is one possible progression of the game:

• Takahashi deletes the first element of A_2. Now we have $A_1 = \left(1, 2, 3\right), A_2 = \left(10\right)$.
• Aoki deletes the last element of A_1. Now we have $A_1 = \left(1, 2\right), A_2 = \left(10\right)$.
• Takahashi deletes the first element of A_1. Now we have $A_1 = \left(2\right), A_2 = \left(10\right)$. The length of every sequence has become 1, so the game ends.

In this case, the sum of the K elements remaining in the end is 12. Note that the players may have made suboptimal moves here.

Sample Input 2

8
1 2
2 1 2
3 1 2 1
4 1 1 1 2
5 1 1 2 2 1
6 2 2 2 2 1 1
7 1 2 1 1 2 2 2
8 2 2 2 1 1 1 1 2

Sample Output 2

12

见题面 Input 部分

见题面 Output 部分