
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
 $P(x)$  is a polynomial with real coefficients such that  $P(a_1) = 0, P(a_{i+1}) = a_i$  ( $i = 1, 2,\ldots$ ) where  $\{a_i\}_{i=1,2,\ldots}$  is an infinite sequence of distinct natural numbers. Determine the possible values of degree of  $P(x)$ .

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