
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
Alice and Bob play a game involving two thousand and eleven \(2011 \times 2011\) grids distributed between them: 1 to Bob and the other 2010 to Alice. Behind closed doors, they fill their grids with the numbers \(1, 2, \ldots, 2011^2\) such that the numbers increase strictly across rows (left-to-right) and down columns (top-to-bottom). No two of Alice's grids can be identical. After filling the grids, Bob can look at Alice's grids and swap numbers on his own grid two at a time, ensuring the grid remains legal (strictly increasing across rows and down columns) after each swap. If a randomly selected grid of Alice's has two integers in the same column in that grid that occur in the same row of Bob's grid, Bob wins. Otherwise, Alice wins. What is the maximum number of swaps Bob may need to ensure victory if he selects his initial grid optimally?

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