
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
$ABCD$, a rectangle with $AB = 15$ and $BC = 20$, is the base of pyramid $P$, which has a height of $30$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let $T$ denote the apex of $P$. If the volume of $P$ is nine times that of $P'$, then find the value of $XT$ and express it as a fraction $\frac{m}{n}$, where $m$ and $n$ are coprime integers. Determine the value of $m + n$.

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