
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) If ∀x<sub>0</sub> ∈ R, the inequality f(x<sub>0</sub>) ≥ |k+3| - |k-2| holds, find the range of values for k.  
(2) If $f(x) ≥ \frac{1}{m} + \frac{1}{n}$ holds for all x ∈ R, find the minimum value of m+n.

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