
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 vectors $\overset{→}{m}=\left(2 \sqrt{3},1\right)$ and $\overset{→}{n}=\left(\cos ^{2} \frac{A}{2}, \sin A\right)$, where $A, B, C$ are the internal angles of $\triangle ABC$.

(1) When $A= \frac{π}{2}$, find the magnitude of $\overset{→}{n}$.

(2) If $C= \frac{2π}{3}$, $|AB|=3$, find the value of $A$ and the length of edge $BC$ when $\overset{→}{m} \cdot \overset{→}{n}$ reaches its maximum value.

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