
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 side length $10$ and a height of $15$. A plane parallel to $ABCD$ slices through $P$, dividing it into a frustum $F$ and a smaller pyramid $P'$. Let $S$ be the center of the frustum's circumsphere and $V$ be the apex of $P$. If the volume of $P$ is nine times that of $P'$, find the length of $SV$.

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