
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
$ABCD$ is a square base of a pyramid $P$ with each side $10$. Pyramid $P$ has a height of $20$. Through pyramid $P$, a plane parallel to square $ABCD$ slices it, forming a frustum $F$ and a smaller pyramid $P'$. Let $X$ represent the center of the circumsphere of frustum $F$, and let $T$ denote the apex of $P$. The volume of $P$ is $15$ times that of $P'$. Find $XT$ and express it in the form $\frac{m}{n}$ where $m$ and $n$ are coprime integers. Calculate $m+n$.

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