
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 the function \( f(x) = a \sqrt{1 - x^2} + \sqrt{1 - x} + \sqrt{1 + x} \). The maximum value of this function is denoted as \( g(a) \). 
(1) Let \( t = \sqrt{1 - x} + \sqrt{1 + x} \). Find the range of values for \( t \), and express \( f(x) \) as a function of \( t \), denoted as \( m(t) \).
(2) Find \( g(a) \).
(3) Find all real numbers \( a \) such that \( g(a) = g\left( \frac{1}{a} \right) \).

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