
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
Boy Vasya wrote the non-zero coefficients of an 8th-degree polynomial \( P(x) \) in a notebook. He then calculated the derivative of the resulting polynomial and noted its non-zero coefficients, continuing in this manner until a constant was obtained, which he also noted.

What is the minimum number of different numbers he could have come up with?

Coefficients are written with their signs, constant terms are also written if there is a term of the form \( \pm x^{n} \), \(\pm 1\) is written.

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