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

1. Let $t = \sqrt{1 + x} + \sqrt{1 - x}$. Express $f(x)$ as a function $m(t)$ of $t$.
2. Find $g(a)$.
3. Determine whether there exists a positive real number $a$ greater than $\sqrt{2}$ such that $g(a) = g(\frac{1}{a})$. If such $a$ exists, find all possible values; otherwise, explain why not.

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