
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 a real number $a < 0$, the function $f(x) = a\sqrt{1 - x^2} + \sqrt{1 + x} + \sqrt{1 - x}$.
- (I) Find the domain of the function $f(x)$;
- (II) Let $t = \sqrt{1 + x} + \sqrt{1 - x}$, find the range of $t$ and express $f(x)$ as a function of $t$, denoted as $h(t)$;
- (III) If the maximum value of the function $f(x)$ is $g(a)$, find $g(a)$.

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