
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 $a < 0$, consider the function 
$$f(x) = a \sqrt{1 - x^2} + \sqrt{1+x} + \sqrt{1-x}.$$  
(1) Let $t = \sqrt{1+x} + \sqrt{1-x}$. Determine the range of values for $t$;  
(2) Express $f(x)$ as a function $h(t)$ of $t$;  
(3) If the maximum value of the function $f(x)$ is represented as $g(a)$, find $g(a)$.

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