
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
A boy named Vasya wrote down the non-zero coefficients of a 9th degree polynomial \( P(x) \) in a notebook. Then, he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, continuing this process until he obtained a constant, which he also wrote down.

What is the minimum number of different numbers that he could have written down?

Coefficients are recorded with their signs, and constant terms are written down as well. If there is a monomial of the form \( \pm x^n \), it is written as \( \pm 1 \).

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