
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 functions $f(x)=\sqrt{1+x}+\sqrt{1-x}$ and $g(x)=\sqrt{1-x^2}$.
$(1)$ Determine the parity of $f(x)$ and find the range of $f(x)$.
$(2)$ Let $F(x)=f(x)+2ag(x)$ where $a<0$. Find the maximum value $h(a)$ of $F(x)$ and then find the minimum value of $h(a)$.

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