
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 $\triangle ABC$ (not including the boundary), 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 $x$, $y$, and $z$ respectively, find the minimum value of $f(x, y, z) = \frac{1}{x} + \frac{4}{y} + \frac{9}{z}$.

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