In acute \(\triangle ABC, \) let \(D, E, F\) be the feet of perpendiculars from \(A, B, C\) to \(BC, CA, AB\) respectively. Line \(EF\) meets the circumcircle of \(\triangle ABC\) at two distinct points \(P\) and \(Q.\) \(DF\) intersects \(BP\) and \(BQ\) at points \(R, S\) respectively. Also, line \(DE\) intersects \(CQ\) and \(CP\) at points \(T, U\) respectively. Find \(\dfrac{AR+AS+AT}{AP+AQ+AU}. \)

This problem is not original. I had fun working on it, so I thought it would be worth sharing.

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