# Any Synthetic Solution?

Geometry Level 5

Triangle $$ABC$$ with circumcircle $$\omega$$ has tangents $$l_1, l_2,$$ and $$l_3$$ at $$A, B,$$ and $$C,$$ respectively.

Let the intersection points of the three tangents be $l_2 \cap l_3 = D, \quad l_1 \cap l_3 = E, \quad l_1 \cap l_2 = F.$ Also, let the point where $$DF$$ and $$AC$$ intersect be $$Q,$$ and let $$EQ \cap BC = S$$ and $$EF \cap BC = T.$$

If $$|AC| = 2|AB|,$$ and $$V$$ is defined as $$V = \dfrac{|ST|}{|BC|},$$ what is $$300V ?$$

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