
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 $f(x)=\ln\left(\frac{3}{{x}^{2}}+a\right)$, where $a\in \mathbb{R}$.<br/>$(1)$ If the domain of $f(x)$ is $\{x|x\neq 0, x\in \mathbb{R}\}$ and the range is $\mathbb{R}$, find the value of $a$;<br/>$(2)$ Under condition (1), when $x_{1}$, $x_{2}\in [c,c+2]$, it always satisfies $|f(x_{1})-f(x_{2})|\leqslant \ln 2$, find the range of $c$.

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