
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
Consider the function $y=a\sqrt{1-x^2} + \sqrt{1+x} + \sqrt{1-x}$ ($a\in\mathbb{R}$), and let $t= \sqrt{1+x} + \sqrt{1-x}$ ($\sqrt{2} \leq t \leq 2$).

(1) Express $y$ as a function of $t$, denoted as $m(t)$.

(2) Let the maximum value of the function $m(t)$ be $g(a)$. Find $g(a)$.

(3) For $a \geq -\sqrt{2}$, find all real values of $a$ that satisfy $g(a) = g\left(\frac{1}{a}\right)$.

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