
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 that $M$ is a point inside $\triangle ABC$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 2\sqrt{3}$, $\angle BAC = 30^{\circ}$, if the areas of $\triangle MBC$, $\triangle MCA$, and $\triangle MAB$ are respectively $\frac{1}{2}$, $x$, and $y$, find the minimum value of $\frac{1}{x} + \frac{4}{y}$.

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