
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 \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that
\[ 
x f(2 f(y)-x) + y^{2} f(2 x-f(y)) = \frac{f(x)^{2}}{x} + f(y f(y)) 
\]
for all \( x, y \in \mathbb{Z} \) such that \( x \neq 0 \).

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