
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 decompositions of an  $8\times 8$  chessboard into  $p$  non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If  $a_i$  is the number of white squares in the  $i$ -th rectangle, then  $a_1<a_2<\ldots <a_p$ . 
Find the maximum value of  $p$  for which such a decomposition is possible. For this value of  $p$ , determine all possible sequences  $a_1,a_2,\ldots ,a_p$ .

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