
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 $f(x)=\log\frac{{1-x}}{{x+1}}$.  
$(1)$ Find the solution set of the inequality $f\left(f\left(x\right)\right)+f\left(\log 3\right) \gt 0$;  
$(2)$ For the function $g(x)=2-a^{x}(a \gt 0,a\neq 1)$, if there exist $x_{1}$ and $x_{2}\in \left[0,1\right)$ such that $f(x_{1})=g(x_{2})$, find the range of real number $a$.

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