
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 hyperbola $S$ with its center at the origin and foci on the x-axis, the eccentricity $e=\frac{\sqrt{6}}{2}$. The minimum distance between a point on the line $\sqrt{3}x-3y+5=0$ and the right focus of the hyperbola $S$ is $\frac{4\sqrt{3}}{3}$.
(1) Find the equation of the hyperbola $S$;
(2) Consider a line passing through the point $(-2, 0)$ with a slope $k$. This line intersects the hyperbola $S$ at points $A$ and $B$. Triangle $ABP$, with $P$ being the point $(0,1)$ and base $AB$, is isosceles. Determine the value of $k$.

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