
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
Divide an $8 \times 8$ chessboard into $p$ rectangles so that:
(1) Each rectangle's sides are along the grid lines of the chessboard;
(2) The number of white squares equals the number of black squares in each rectangle;
(3) For the $i$-th rectangle, let the number of white squares be $a_i$, and $a_1 < a_2 < \cdots < a_p$.
Determine the maximum value of $p$ and list all possible sequences $a_1, a_2, \cdots, a_p$ for this maximum $p$.

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