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

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