
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 $f(x) = |2x - 1| + a|x - 1|$,
(I) Solve the inequality $f(x) \geq 4$ for $x$ when $a = 1$.
(II) If the solution set of $f(x) \geq |x - 2|$ includes the interval $\left[\frac{1}{2}, 2\right]$, find the range of the real number $a$.

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