
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 $AB = 10$ and $BC = 15$, is the base of pyramid $Q$, which has a height of 30. A plane parallel to $ABCD$ is passed through $Q$, dividing $Q$ into a frustum $G$ and a smaller pyramid $Q'$. Let $Y$ be the center of the circumsphere of $G$, and $U$ be the apex of $Q$. If the volume of $Q$ is nine times that of $Q'$, find the length of $YU$ expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

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