
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 functions $f(x) = x^2 - 1$ and $g(x) = a|x - 1|$,  
(1) When $a = 1$, solve the equation $|f(x)| = g(x)$;  
(2) If the equation $|f(x)| = g(x)$ has two distinct solutions, find the value of $a$;  
(3) If for $x \in \mathbb{R}$, the inequality $f(x) \geq g(x)$ always holds, find the range of the real number $a$.

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