
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
For each of the following equations, determine the set of polynomials \( P \in \mathbb{K}[X] \) that satisfy the condition.

1. \( P \circ P = P \)
2. \( P\left(X^{2}\right)=\left(X^{2}+1\right) P(X) \)
3. \( P(2X) = P(X) - 1 \)
4. \( 16 P\left(X^{2}\right) = P(2X)^{2} \)

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