
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
Given that the domain of function $f(x)$ is $(-1,1)$, for any $x, y \in (-1,1)$, $f(x) + f(y) = f(\frac{x + y}{1 + xy})$. Also, when $x < 0$, $f(x) > 0$.
(1) Verify if the function $f(x) = \log \frac{1 - x}{1 + x}$ satisfies these conditions;
(2) If $f(\frac{a + b}{1 + ab}) = 1$, $f(\frac{a - b}{1 - ab}) = 2$, and $|a| < 1$, $|b| < 1$, find the values of $f(a)$ and $f(b)$;
(3) If $f(-\frac{1}{2}) = 1$, solve the equation $f(x) = -\frac{1}{2}$ for $x$.

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