
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
Given the function $f(x) = \sin(2x + \frac{\pi}{3}) + \cos(2x + \frac{\pi}{6}) + m\sin(2x)$ ($m \in \mathbb{R}$), and $f(\frac{\pi}{12}) = 2$.

(Ⅰ) Find the value of $m$;  
(Ⅱ) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $b=2$, $f(\frac{B}{2}) = \sqrt{3}$, and the area of $\triangle ABC$ is $\sqrt{3}$, find the perimeter of $\triangle ABC$.

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