
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 $f(x) = \sqrt{|2x-1|+|x+1|-a}$ with a domain of $R$.

(1) Find the range of real number values for $a$.
(2) If the maximum value of $a$ is $k$, and $m + n = 2k$ ($m > 0$, $n > 0$), find the minimum value of $\frac{1}{m} + \frac{4}{n}$.

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