
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) = |x + m| + |2x - 1| (m > 0).
(1) When m = 1, solve the inequality f(x) ≥ 3;
(2) When x ∈ [m, 2m²], the inequality $\frac{1}{2}$f(x) ≤ |x + 1| always holds true. Find the range of the real number m.

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