
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 functions $f(x)=\log_a(x+m)$ and $g(x)=\log_a(1-x)$. If the zero of the function $F(x)=f(x)-g(x)$ is $0$.

$(1)$ Find the value of $m$ and the domain of the function $F(x)$;

$(2)$ Determine the parity of $F(x)$ and explain the reason;

$(3)$ Find the range of $x$ for which $F(x) > 0$ holds.

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