
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 $ABC$ with side length $1$, point $E$ is on $AC$, and $\overrightarrow{AC}=4\overrightarrow{AE}$. Point $P$ is on $BE$, and it satisfies $\overrightarrow{AP}=m\overrightarrow{AB}+n\overrightarrow{AC}$ ($m > 0,n > 0$). Find the minimum value of $\frac{1}{m}+\frac{1}{n}$, and when this minimum value is achieved, calculate $\left| \overrightarrow{AP} \right|=$ ______.

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