
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 triangle $\triangle ABC$, point $E$ is on line segment $AC$, and $\overrightarrow {AC}=4 \overrightarrow {AE}$. Point $P$ is on line segment $BE$, and $\overrightarrow {AP}=m \overrightarrow {AB}+n \overrightarrow {AC}$ ($m>0$, $n>0$). Determine the magnitude of vector $\overrightarrow {a}=(m,n)$ when $\frac {1}{m}+ \frac {1}{n}$ takes its minimum value.

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