
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 $AD=8$, $BE=10$, $CF=20$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, inclined at an angle of 45 degrees and not intersecting the triangle. Points $D$, $E$, and $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, determine the value of $x$.

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