
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
A pyramid with a square base $ABCD$ where $AB = 10$ and height $20$ is cut by a plane parallel to its base, creating a smaller pyramid and a frustum. Let $X$ be the center of the circumsphere of the frustum and $T$ the apex of the pyramid. If the volume of the original pyramid is nine times that of the smaller pyramid, find the distance $XT$ as a simplified fraction and compute the sum of its numerator and denominator.

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