
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 angles $A$, $B$, and $C$ are the internal angles of $\triangle ABC$, and $a$, $b$, and $c$ are the lengths of their respective opposite sides. The vectors $m=(2\sqrt{3}\sin{\frac{A}{2}}, \cos^2{\frac{A}{2}})$ and $n=(\cos{\frac{A}{2}}, -2)$ are given, with $m \perp n$.

(1) Find the measure of angle $A$.
(2) If $a=2$ and $\cos{B}=\frac{\sqrt{3}}{3}$, find the length of $b$.

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