
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 \(\omega\) denote the incircle of triangle \(ABC\). The segments \(BC, CA\), and \(AB\) are tangent to \(\omega\) at \(D, E\), and \(F\), respectively. Point \(P\) lies on \(EF\) such that segment \(PD\) is perpendicular to \(BC\). The line \(AP\) intersects \(BC\) at \(Q\). The circles \(\omega_1\) and \(\omega_2\) pass through \(B\) and \(C\), respectively, and are tangent to \(AQ\) at \(Q\); the former meets \(AB\) again at \(X\), and the latter meets \(AC\) again at \(Y\). The line \(XY\) intersects \(BC\) at \(Z\). Given that \(AB=15\), \(BC=14\), and \(CA=13\), find \(\lfloor XZ \cdot YZ \rfloor\).

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