
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
Suppose  $P$  is a quintic polynomial with real coefficients with  $P(0)=2$  and  $P(1)=3$  such that  $|z|=1$  whenever  $z$  is a complex number satisfying  $P(z) = 0$ .  What is the smallest possible value of  $P(2)$  over all such polynomials  $P$ ?

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