
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
Cut an $8 \times 8$ chessboard into $p$ rectangles, without damaging any individual square. The cutting must satisfy the following conditions:
1. Each rectangle must contain an equal number of black and white squares.
2. Let $a_i$ be the number of white squares in the $i$-th rectangle, then $a_1 < a_2 < \cdots < a_p$.

Determine the maximum value of $p$ that allows such a cutting method to exist, and for this value of $p$, find all possible values of $a_1, a_2, \cdots, a_p$.

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