
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 acute $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. Given vectors $\overrightarrow{m} = \left( \frac{1}{2}, \cos A \right)$ and $\overrightarrow{n} = \left( \sin A, -\frac{\sqrt{3}}{2} \right)$, and $\overrightarrow{m} \perp \overrightarrow{n}$  
$(1)$ Find the magnitude of angle $A$;  
$(2)$ If $a=7$ and $b=8$, find the area of $\triangle ABC$.

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