
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 $f(x)=m\sqrt{lnx-\frac{1}{4}}+2x+\frac{1}{2}n$ has a root in the interval $\left[2,4\right]$, then the minimum value of $\sqrt{{m^2}+{n^2}}$ is ____.

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