
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
Avant-garde artist Zmiy Kletockin painted several cells of a $7 \times 7$ board following the rule: each subsequently painted cell must be adjacent by side to the previously painted cell but must not be adjacent to any other previously painted cell. He managed to paint 31 cells.

Break his record by painting a) 32 cells; b) 33 cells.

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