
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, where $c > b$. Given that $\overrightarrow{AB} \cdot \overrightarrow{AC} = 20$ and the area of $\triangle ABC$ is $10\sqrt{3}$. Find:

$(1)$ The measure of angle $\angle A$.

$(2)$ Let point $O$ be the circumcenter of $\triangle ABC$, and satisfy $\overrightarrow{OB} \cdot \overrightarrow{OC} = -\frac{49}{6}$. Find the length of side $a$.

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