
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
On an $8 \times 8$ chessboard, we wrote numbers on each square, placing a 0 in every corner. Each number on the board is not greater than the average of the numbers in the squares directly adjacent to it. What numbers did we write on the chessboard?

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