
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 the function $f(x)=\lg \frac{1+x}{1+ax} (a\ne 1)$ is an odd function,

$(1)$ Find the value of $a$;

$(2)$ If $g(x)=f(x)+\frac{2}{1+{{2}^{x}}}, x\in (-1,1)$, find the value of $g(\frac{1}{2})+g(-\frac{1}{2})$.

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