
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
A black bishop and a white king are placed randomly on a  $2000 \times 2000$  chessboard (in distinct squares). Let  $p$  be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then  $p$  can be expressed in the form  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Compute  $m$ .

*Proposed by Ahaan Rungta*

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