
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 $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given $\overrightarrow{m} = (\sin^2 \frac{B+C}{2}, 1)$, $\overrightarrow{n} = (-2, \cos 2A + 1)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

(Ⅰ) Find the degree measure of angle $A$;  
(Ⅱ) When $a=2\sqrt{3}$, and the area $S$ of $\triangle ABC$ is $S= \frac{a^2+b^2-c^2}{4\sqrt{3}}$, find the value of side $c$ and the area of $\triangle ABC$.

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