
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 $f(x) = |x-1| - 2|x+1|$, the maximum value of $f(x)$ is $a$.
(1) Find the value of $a$.
(2) If $\frac{1}{m} + \frac{1}{2n} = a$ where $m>0$ and $n>0$, compare the magnitude of $2m+n$ with $2$.

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