
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 a function $f(x)$ with domain $(-1,1)$, for any $x,y \in (-1,1)$, it holds that $f(x) + f(y) = f\left( \frac{x+y}{1+xy} \right)$. Additionally, when $x<0$, $f(x) > 0$.
1. Verify whether the function $f(x) = \lg \frac{1-x}{1+x}$ satisfies these conditions.
2. If $f\left( \frac{a+b}{1+ab} \right) = 1$ and $f\left( \frac{a-b}{1-ab} \right) = 2$ with $|a| < 1$ and $|b| < 1$, find the values of $f(a)$ and $f(b)$.
3. If $f\left( -\frac{1}{2} \right) = 1$, solve the equation $f(x) = -\frac{1}{2}$ for $x$.

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