In triangle $ABC,$ $\overline{CD}$ is the bisector of angle $C,$ with $D$ on $\overline{AB}.$  If $\cos \frac{C}{2} = \frac{1}{3}$ and $CD = 6,$ compute
\[\frac{1}{BC} + \frac{1}{AC}.\]
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In triangle $ABC,$ $\overline{CD}$ is the bisector of angle $C,$ with $D$ on $\overline{AB}.$  If $\cos \frac{C}{2} = \frac{1}{3}$ and $CD = 6,$ compute
\[\frac{1}{BC} + \frac{1}{AC}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1}{9}
In triangle $ABC,$ $\overline{CD}$ is the bisector of angle $C,$ with $D$ on $\overline{AB}.$  If $\cos \frac{C}{2} = \frac{1}{3}$ and $CD = 6,$ compute
\[\frac{1}{BC} + \frac{1}{AC}.\]
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