
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
Tatjana imagined a polynomial  $P(x)$  with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer  $k$  and Tatjana tells her the value of  $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.

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