
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 real-valued functions \( f \) defined on the set of all non-zero real numbers such that:

1. \( f(1)=1 \),
2. \( f\left( \frac{1}{x+y} \right)=f\left( \frac{1}{x} \right)+f\left( \frac{1}{y} \right) \) for all non-zero \( x, y \) such that \( x+y \neq 0 \),
3. \( (x+y) f(x+y) = x y f(x) f(y) \) for all non-zero \( x, y \) such that \( x+y \neq 0 \).

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