
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
Let \( D \) be the set of real numbers excluding -1. Find all functions \( f: D \rightarrow D \) such that for all \( x, y \in D \) satisfying \( x \neq 0 \) and \( y \neq -x \), the equality

\[ (f(f(x))+y) f\left(\frac{y}{x}\right) + f(f(y)) = x \]

holds.

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