
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 $n \times n$ grid, fill it with integers from 1 to $n^2$ according to the following rule: Place 1 in any cell, the row of 2 should match the column of 1, the row of 3 should match the column of 2, and so on. What is the difference between the sum of all numbers in the row where 1 is placed and the sum of all numbers in the column where $n^2$ is placed?

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