
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 that the angle between vectors \(\boldsymbol{m}\) and \(\boldsymbol{n}\) is \(\frac{\pi}{6}\), and that \(|\boldsymbol{m}| = \sqrt{3}\) and \(|\boldsymbol{n}| = 2\). In \(\triangle ABC\), \(\overrightarrow{A B} = 2 \boldsymbol{m} + 2 \boldsymbol{n}\) and \(\overrightarrow{A C} = 2 \boldsymbol{m} - 6 \boldsymbol{n}\), with \(D\) being the midpoint of side \(B C\), find the length \(|\overrightarrow{A D}|\).

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