
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $n\geq 3$  be an odd number. We will play a game using a  $n$  by  $n$  grid. The game is comprised of  $n^2$  turns, in every turn, we will perform the following operation sequentially. $\bullet$  We will choose a square with an unwritten integer, and write down an integer among 1 through  $n^2$ . We can write down any integer only at once through the game. $\bullet$  For each row, colum including the square, if the sum of integers is a multiple of  $n$ , then we will get 1 point (both of each sum is a multiple of  $n$ , we will get 2 points).

Determine the maximum possible value of the points as the total sum that we can obtain by the end of the game.

Present the answer in LaTex format: \boxed{Your answer}
user