
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 the function $f(x)$ have the domain $(-\infty,+\infty)$ and satisfy the condition: there exist $x_1 \neq x_2$ such that $f(x_1) \neq f(x_2)$. For all $x$ and $y$, the equation $f(x+y) = f(x) \cdot f(y)$ holds. Find: (1) $f(0)$; (2) Determine the sign of $f(x)$ for any value of $x$.

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