
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
In an equilateral triangle $\triangle ABC$ with side length $1$, $E$ is a point on $AC$ such that $\overrightarrow{AC} = 4\overrightarrow{AE}$, and $P$ is a point on $BE$ such that $\overrightarrow{AP} = m\overrightarrow{AB} + n\overrightarrow{AC}$ ($m > 0$, $n > 0$), then the minimum value of $\frac{1}{m} + \frac{1}{n}$ is attained when $|\overrightarrow{AP}| =$ $\_\_\_\_\_\_\_\_\_$.

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