
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 $f(x) = |x-b| + |x+b|$.  
(1) When $b=1$, find the solution set for $f(x) \leq x+2$;  
(2) When $x=1$, if the inequality $f(x) \geq \frac{|a+1|-|2a-1|}{|a|}$ holds for any real number $a \neq 0$, find the range of the real number $b$.

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