
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
The function \( f: \mathbf{R} \rightarrow \mathbf{R} \) satisfies the identity
\[ 
f(x y) \equiv \frac{f(x) + f(y)}{x + y}, \quad x, y \in \mathbf{R}, \quad x + y \neq 0 
\]
Does there exist a value \( x \in \mathbb{R} \) for which \( f(x) \neq 0 \)?

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