
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

Petya chose a natural number \( a > 1 \) and wrote fifteen numbers on the board: \( 1 + a, 1 + a^2, 1 + a^3, \ldots, 1 + a^{15} \). Then he erased several numbers so that any two remaining numbers are coprime. What is the maximum number of numbers that could have remained on the board?

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