The function $f(x)$ satisfies
\[f(x) + f(2x) + f(2 - x) + f(x + 1) = x\]for all real numbers $x.$  Find $f(0).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The function $f(x)$ satisfies
\[f(x) + f(2x) + f(2 - x) + f(x + 1) = x\]for all real numbers $x.$  Find $f(0).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-\frac{1}{4}
The function $f(x)$ satisfies
\[f(x) + f(2x) + f(2 - x) + f(x + 1) = x\]for all real numbers $x.$  Find $f(0).$
train