
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 $ABC$, it is given that $\overrightarrow {AB}\cdot \overrightarrow {AC}=2\sqrt {3}$, and $\angle BAC=30^\circ$. For a point $M$ inside the triangle $ABC$, define $f(M)=(x,y,z)$ where $x$, $y$, and $z$ respectively represent the areas of triangles $MBC$, $MAC$, and $MAB$. If $f(M)=(x,y,\frac{1}{2})$, then find the minimum value of $\frac{1}{x}+\frac{4}{y}$.

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