
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 $\triangle ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, consider vectors $\vec{m} = (\cos A, \cos B)$ and $\vec{n} = (b+2c, a)$, and it is known that $\vec{m} \perp \vec{n}$.
1. Find the magnitude of angle $A$;
2. If $a=4\sqrt{3}$ and $b+c=8$, find the height $h$ from vertex $A$ onto side $BC$.

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