
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 $f(x) = |2x-1|$.
(1) Find the solution set of the inequality $f(x) + |x+1| < 2$;
(2) If the function $g(x) = f(x) + f(x-1)$ has a minimum value of $a$, and $m+n=a$ ($m>0$, $n>0$), find the minimum value of $\frac{4}{m} + \frac{1}{n}$.

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