
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 $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, let vectors $\overrightarrow{m}=(\cos B,\sin C)$ and $\overrightarrow{n}=(\cos C,-\sin B)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=\frac{1}{2}$.

$(1)$ Find the magnitude of the interior angle $A$;

$(2)$ If the radius of the circumcircle of $\triangle ABC$ is $R=2$, find the maximum value of the perimeter $L$ of $\triangle ABC$.

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