
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 nonzero coefficients of a tenth-degree polynomial \( P(x) \) in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, and continued this process until he arrived at a constant, which he also wrote down.

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

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

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