
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
Find all polynomials  $P(x)$  with real coefficients such that for all pairwise distinct positive integers  $x, y, z, t$  with  $x^2 + y^2 + z^2 = 2t^2$  and  $\gcd(x, y, z, t ) = 1,$  the following equality holds
\[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\]**Note.**  $P^2(k)=\left( P(k) \right)^2.$ 

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