
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
Consider a partition of an $8 \times 8$ chessboard into $p$ rectangles whose interiors are disjoint such that each of them has an equal number of white and black cells. Assume that $a_{1}<a_{2}<\cdots<a_{p}$, where $a_{i}$ denotes the number of white cells in the $i$-th rectangle. Find the maximal $p$ for which such a partition is possible, and for that $p$, determine all possible corresponding sequences $a_{1}, a_{2}, \ldots, a_{p}$.

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