
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
Given the function \( f(x) = ax^2 - 1 \) where \( a \in \mathbb{R} \) and \( x \in \mathbb{R} \), and sets \( A = \{ x \mid f(x) = x \} \) and \( B = \{ x \mid f(f(x)) = x \} \), such that \( A = B \neq \emptyset \), find the range of the real number \( a \).

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