
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
Let \( ABC \) be an equilateral triangle with side length 8. Let \( X \) be on side \( AB \) so that \( AX = 5 \) and \( Y \) be on side \( AC \) so that \( AY = 3 \). Let \( Z \) be on side \( BC \) so that \( AZ, BY, CX \) are concurrent. Let \( ZX, ZY \) intersect the circumcircle of \( AXY \) again at \( P, Q \) respectively. Let \( XQ \) and \( YP \) intersect at \( K \). Compute \( KX \cdot KQ \).

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