
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 $f(x)=\ln (1+x)-\ln (1-x)$.
$(1)$ Discuss the monotonicity of the function $F(x)=f(x)+ax(a\neq 0)$;
$(2)$ If $f(x) > k(x^{3}-3x)$ holds for $x\in(0,1)$, find the range of $k$.

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