
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
Segments $AD = 15$, $BE = 9$, and $CF = 27$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $PT$, not intersecting the triangle. Points $D$, $E$, and $F$ are the intersection points of $PT$ with the perpendiculars. If triangle $ABC$ is a right triangle at $C$, and $x$ is the length of the perpendicular segment $GH$ drawn to $PT$ from the centroid $G$ of the triangle, find $x$.
A) $\frac{51}{3}$
B) $17$
C) $\frac{56}{3}$
D) $21$
E) $30$

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