
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 $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m}=(-\cos B, \sin C)$ and $\overrightarrow{n}=(-\cos C, -\sin B)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=\frac{1}{2}$.

(Ⅰ) Find the magnitude of angle $A$;  
(Ⅱ) If $b+c=4$ and the area of $\triangle ABC$, $S=\sqrt{3}$, find the value of $a$.

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