
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 $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively. Given the vectors $\vec{m} = (\cos^2 \frac{A}{2}, \cos B)$, $\vec{n} = (-a, 4c+2b)$, and $\vec{p} = (1, 0)$, and that $(\vec{m} - \frac{1}{2}\vec{p})$ is parallel to $\vec{n}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $a = \sqrt{3}$, find the maximum perimeter of $\triangle ABC$.

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