
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
An $8 \times 8$ chessboard is divided into $p$ rectangles. The division must follow these rules:
1. Each rectangle contains the same number of white and black squares.
2. If the $i$-th rectangle contains $a_{i}$ white squares, then $a_{1} < a_{2} < \cdots < a_{p}$.

Find the maximum possible value of $p$, and for this maximum value, list all possible sequences $a_{1}, a_{2}, \cdots, a_{p}$.

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