
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

Given an odd number \( n \geqslant 3 \), a game is played on an \( n \times n \) grid where \( n^{2} \) steps are performed. Each step follows these rules:

1. Choose an empty cell and fill it with a positive integer not less than 1 and not greater than \( n^{2} \), with each integer being used exactly once throughout the game.
2. If the sum of the numbers in the row or the column of the newly filled cell is a multiple of \( n \), the player scores 1 point. If both the row and column sums are multiples of \( n \), the player scores 2 points.

Find the highest possible score one can achieve in this game.

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