
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 that  $P(x)$  is a polynomial with the property that there exists another polynomial  $Q(x)$  to satisfy  $P(x)Q(x)=P(x^2)$ .  $P(x)$  and  $Q(x)$  may have complex coefficients. If  $P(x)$  is a quintic with distinct complex roots  $r_1,\dots,r_5$ , find all possible values of  $|r_1|+\dots+|r_5|$ .

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