Trigonometry · Superchallenge problem

Two surveyors stand at points $A$ and $B$, $60$ m apart, on one bank of a river. Two trees $P$ and $Q$ stand on the far bank (the same side of line $AB$). The surveyors measure $\angle BAP = 105^\circ$, $\angle ABP = 45^\circ$, $\angle BAQ = 45^\circ$, and $\angle ABQ = 105^\circ$. Find the distance $PQ$, in metres, in simplest radical form.