1984 USAMO 第 3 题

$P$, $A$, $B$, $C$, and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.

$P$、$A$、$B$、$C$ 和 $D$ 是空间中的五个不同点，使得 $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$，其中 $\theta$ 是给定的锐角。确定$\angle APC + \angle BPD$ 的最大值和最小值。