Tangent Circles Make Great Triangles!

Geometry Level 3

Let \(\omega_1\) and \(\omega_2\) be two externally tangent circles at \(T\) with \(\omega_1\) smaller than \(\omega_2\). Let their common external tangents intersect at \(S\). Let a line through \(S\) intersect circles \(\omega_1\) and \(\omega_2\) at \(A,B,C,D\) in that order. Given that \(TC = 3\) and \(TA = 4\), find \(AC\).

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