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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