
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
Let \( P(x) \) be a polynomial with leading coefficient 1, and let the sequence of integers \( a_1, a_2, a_3, \ldots \) be such that \( P(a_1) = 0 \), \( P(a_2) = a_1 \), \( P(a_3) = a_2 \), and so on. The numbers in the sequence do not repeat. What degree can \( P(x) \) have?

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