
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
We define the bulldozer of triangle  $ABC$  as the segment between points  $P$  and  $Q$ , distinct points in the plane of  $ABC$  such that  $PA\cdot BC=PB\cdot CA=PC\cdot AB$  and  $QA\cdot BC=QB\cdot CA=QC\cdot AB$ . Let  $XY$  be a segment of unit length in a plane  $\mathcal{P}$ , and let  $\mathcal{S}$  be the region of  $\mathcal P$  that the bulldozer of  $XYZ$  sweeps through as  $Z$  varies across the points in  $\mathcal{P}$  satisfying  $XZ=2YZ$ . Find the greatest integer that is less than  $100$  times the area of  $\mathcal S$ .

*Proposed by Michael Ren*

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