
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 $f\left(x\right)=\sqrt{1+x}+\sqrt{1-x}$,  
$(1)$ Find the domain and range of the function $f\left(x\right)$;  
$(2)$ Let $F\left(x\right)=m\sqrt{1-{x}^{2}}+f\left(x\right)$, find the expression $g\left(m\right)$ for the maximum value of the function $F\left(x\right)$.

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