
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 two plane vectors $m$ and $n$ with an angle of $\frac{\pi}{6}$ between them, and $|m|=\sqrt{3}$, $|n|=2$. In $\triangle ABC$, $\overrightarrow{AB}=2m+n$, $\overrightarrow{AC}=2m-6n$. If $D$ is the midpoint of $BC$, then $|\overrightarrow{AD}|=$ ______.

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