
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 $f(x) = |2x+1| - |x| - 2$.
(1) Solve the inequality $f(x) \geq 0$;
(2) If there exists a real number $x$ such that $f(x) - a \leq |x|$, find the minimum value of the real number $a$.

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