
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 with two thousand and eleven $2011 \times 2011$ grids distributed between them—1 grid to Bob and 2010 grids to Alice. They fill their grids with the numbers $1, 2, \ldots, 2011^2$ so that the numbers across rows (left-to-right) and down columns (top-to-bottom) are strictly increasing. Each of Alice's grids must be filled uniquely. After filling, Bob can look at Alice's grids and swap numbers on his own grid, maintaining the numerical order. When he finishes swapping, a grid of Alice's is selected randomly. If two integers in the same column of Alice's selected grid appear in the same row of Bob's grid, Bob wins. Otherwise, Alice wins. If Bob chooses his grid optimally, what is the maximum number of swaps Bob may need to guarantee victory?

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