
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 $ABC$, denote the sides opposite angles $A$, $B$, $C$ as $a$, $b$, $c$ respectively. Angle $A$ is an acute angle. Let vector $\overrightarrow{m} = (\cos A, \sin A)$ and vector $\overrightarrow{n} = (\cos A, -\sin A)$. Given that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{1}{2}$,  
(I) Find the magnitude of angle $A$ and the angle between vectors $\overrightarrow{m}$ and $\overrightarrow{n}$.  
(II) If $a = \sqrt{5}$, find the maximum area of triangle $ABC$.

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