
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 $f(x) = |x-a|$.  
(I) If $a=1$, solve the inequality: $f(x) \geq 4-|x-1|$.  
(II) 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}
user