
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 $f(x)=a^{x}-a^{-x}$ and $g(x)=a^{x}+a^{-x}$, where $a>0$ and $a\neq 1$.
$(1)$ If $g(2x+1)\leq g(x+2)$, find the range of $x$.
$(2)$ Let $F(x)=\frac{f(x)}{g(x)}$. If $\forall x_{1}, x_{2} \in [\frac{1}{2},1]$, it holds that $\frac{F(2x_{1})}{F(x_{1})} \geq \frac{18}{17} \times \frac{g(4x_{2})}{g(2x_{2})}$, find the range of $a$.

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