
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 $f(x)=\sqrt{2+x}-\sqrt{2-x}$.  
$(1)$ Find the domain and range of $f(x)$;  
$(2)$ Let $F(x)=\left[f(x)\right]^{2}-2af(x)+1$, and denote the minimum value of $F(x)$ as $g(a)$. Find the maximum value of $g(a)$.

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