
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 $a\geq0$, and consider the function $f(x)=a\sqrt{1-x^2}+\sqrt{1+x}-\sqrt{1-x}$. The maximum value of $f(x)$ is denoted as $g(a)$.
(1) Define $t=\sqrt{1+x}-\sqrt{1-x}$. Determine the range of $t$ and express $f(x)$ as a function of $t$, denoted as $m(t)$.
(2) Find $g(a)$.
(3) Find all real numbers $a$ satisfying $g(a)=g\left(\frac{1}{a}\right)$.

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