Let $P(x)$ be a polynomial such that
\[P(P(x)) + P(x) = 6x\]for all real numbers $x.$  Find the sum of all possible values of $P(10).$
Question:
Let $P(x)$ be a polynomial such that
\[P(P(x)) + P(x) = 6x\]for all real numbers $x.$  Find the sum of all possible values of $P(10).$
Answer:
Let's think step by step.

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Let $P(x)$ be a polynomial such that
\[P(P(x)) + P(x) = 6x\]for all real numbers $x.$  Find the sum of all possible values of $P(10).$
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