
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 $f(x) = |x - a|$.

(1) If $a = 1$, solve the inequality: $f(x) \geq 4 - |x - 1|$;

(2) If the solution set of $f(x) \leq 1$ is $[0, 2]$, and $\frac{1}{m} + \frac{1}{2n} = a (m > 0, n > 0)$, find the minimum value of $mn$.

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