
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
Triangle $ABC$ has vertices $A$, $B$, $C$ with perpendicular segments $AD = 15$, $BE = 8$, and $CF = 30$ drawn to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, and $F$ are intersections with $RS$. A perpendicular segment $GH$ is then drawn from the centroid $G$ of triangle $ABC$ to $RS$. Find the length of $GH$.
A) $\frac{43}{3}$
B) $16$
C) $\frac{53}{3}$
D) $\frac{82}{3}$
E) Undetermined

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