In triangle \(ABC,\) it is given that angles \(B\) and \(C\) are congruent. Points \(P\) and \(Q\) lie on \(\overline{AC}\) and \(\overline{AB},\) respectively, so that \(AP = PQ = QB = BC.\) Angle \(ACB\) is \(r\) times as large as angle \(APQ,\) where \(r\) is a positive real number. Find the greatest integer that does not exceed \(1000r\).
(第十八届AIME1 2000 第14题)