
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 $\triangle ABC$ with $M$ as the midpoint of side $BC$, $AN$ bisecting $\angle BAC$ and side $AC$, and $BN\perp AN$, find the length of $MN$ given that the sides $AB$ and $AC$ have lengths $15$ and $20$ respectively.

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