
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

On the extension of edge \( ST \) beyond point \( T \) of a regular quadrilateral pyramid \( SPQRT \) with vertex \( S \), point \( B \) is taken such that the distance from it to the plane \( SPQ \) is \( \frac{9 \sqrt{7}}{2} \). Find the segment \( BT \) if \( QR = 12 \) and \( SR = 10 \).

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