If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ is on $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, find the ratio of the length of $PQ$ to the length of $AQ$. The original answer is in \(\frac{k\sqrt{3}}{m}\) format, please give the value of k + m.

Let's think step by step and output the final answer within \boxed{}.
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