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 $P(10).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
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 $P(10).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-10
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 $P(10).$
train