
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 $f(x)=\ln\left(\frac{2}{x}+a\right)$, where $a\in\mathbb{R}$.  
$(1)$ If the function $F(x)=f(x)-\ln\left[(2-a)x+3a-3\right]$ has a unique zero, find the range of real numbers for $a$.  
$(2)$ If for any real number $m\in\left[\frac{3}{4},1\right]$, and for any $x_{1},x_{2}\in \left[m,4m-1\right]$, it holds that $|f(x_{1})-f(x_{2})|\leqslant \ln 2$, find the range of positive real numbers for $a$.

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