
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
For the function \(f(x)\), if \(f(x) = x\), then \(x\) is called the "fixed point" of \(f(x)\); if \(f(f(x)) = x\), then \(x\) is called the "stable point" of \(f(x)\). The sets of the "fixed points" and the "stable points" of the function \(f(x)\) are denoted by \(A\) and \(B\), respectively. Given that \(f(x) = a x^{2} - 1\) (where \(a \in \mathbb{R}\), \(x \in \mathbb{R}\)) and \(A = B \neq \varnothing\), find the range of the real number \(a\).

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