
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 points $P(\sqrt{3}, 1)$, $Q(\cos x, \sin x)$, and $O$ as the origin of coordinates, the function $f(x) = \overrightarrow{OP} \cdot \overrightarrow{QP}$

(Ⅰ) Find the smallest positive period of the function $f(x)$;

(Ⅱ) If $A$ is an internal angle of $\triangle ABC$, $f(A) = 4$, $BC = 3$, and the area of $\triangle ABC$ is $\frac{3\sqrt{3}}{4}$, find the perimeter of $\triangle ABC$.

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