
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
Let $a$ be a real number, and let the maximum value of the function $f(x) = a\sqrt{1 - x^2} + \sqrt{1 + x} + \sqrt{1 - x}$ be $g(a)$.

(1) Let $t = \sqrt{1 + x} + \sqrt{1 - x}$. Determine the range of $t$ and express $f(x)$ as a function of $t$, denoted by $m(t)$;

(2) Find $g(a)$;

(3) Find all real values of $a$ that satisfy $g(a) = g(\frac{1}{a})$.

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