
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 such that no single square is cut in half. Each such division must satisfy the following conditions:

(a) Each rectangle must contain an equal number of white and black squares.

(b) If $a_{i}$ denotes the number of white squares in the $i$-th rectangle, the inequality $a_{1} < a_{2} < \ldots < a_{p}$ must hold.

Determine the largest possible value of $p$ for which such a division exists.

Additionally, generate the sequences $a_{1}, a_{2}, \ldots, a_{p}$ for this value of $p$.

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