
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 $f(x) = \log_{a}(1-x) + 3$, where $(a > 0$ and $a \neq 1)$, the graph passes through the point $P(-2,4)$. The function $g(x) = b - \frac{2}{{3}^{x} + 1}$ is an odd function.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ Find the zeros of the function $F(x) = g(x) + 3^{x} - 2$.  
$(3)$ If the inequality in terms of $x$, $m + \log_{3}(\frac{1+x}{1-x}) < f(x)$ holds true on the interval $(-1,0)$, find the range of positive real numbers for $m$.

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