
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
Consider a polynomial  $P(x) \in \mathbb{R}[x]$ , with degree  $2023$ , such that  $P(\sin^2(x))+P(\cos^2(x)) =1$  for all  $x \in \mathbb{R}$ . If the sum of all roots of  $P$  is equal to  $\dfrac{p}{q}$  with  $p, q$  coprime, then what is the product  $pq$ ?

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