
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
Given  $a, \theta \in \mathbb R, m \in \mathbb N$ , and  $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$ , factorize  $P(x)$  as a product of  $m$  real quadratic polynomials.

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