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

What is the smallest number of distinct numbers he might have obtained?

The coefficients are recorded with their signs; if there are free terms, they are also written down. If there's a monomial of the form \( \pm x^{n} \), \( \pm 1 \) is written.

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