
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 function \( f(x) = a \sqrt{1-x^2} + \sqrt{1+x} + \sqrt{1-x} \) have its maximum value denoted as \( g(a) \).

1. Let \( t = \sqrt{1+x} + \sqrt{1-x} \); find the range of \( t \) and express \( f(x) \) as a function \( m(t) \) in terms of \( t \).
2. Find \( g(a) \).
3. Determine all real numbers \( a \) such that \( g(a) = g\left(\frac{1}{a}\right) \).

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