Suppose there exists a triangle \(\triangle{ABC}\) where \(P\) and \(Q\) are points on segments \(\overline{AB}\) and \(\overline{AC}\) respectively, such that:

(1) \(\overline{AP} = \overline{AQ}\).

Let \(S\) and \(R\) be distinct points on segment \(\overline{BC}\) such that:

(2) \(S\) lies between points \(B\) and \(R\)

(3) \(\angle{BPS} = \angle{PRS}\), and \(\angle{CQR} = \angle{QSR}\).

Prove that points \(P,Q,R,S\) are concyclic points.

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