
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(\cos A,\cos B)$ and $\overrightarrow{n}=(b+2c,a)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

(1) Find the measure of angle $A$.
(2) If $a=4 \sqrt {3}$ and $b+c=8$, find the length of the altitude $h$ on edge $AC$.

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