
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 a point M inside triangle $\triangle ABC$, $\vec{AB} \cdot \vec{AC} = 2\sqrt{3}$, $\angle BAC = 30^\circ$. Define $f(x) = (m, n, p)$ where $m$, $n$, and $p$ represent the areas of $\triangle MBC$, $\triangle MAC$, and $\triangle MAB$, respectively. If $f(Q) = \left(\frac{1}{2}, x, y\right)$ and $\frac{1}{x} + \frac{4}{y} = a$, find the range of the value $\frac{a^{2}+2}{a}$.

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