Let $\omega = e^{2 \pi i/1729}.$  Compute
\[\sum_{k = 1}^{1728} \frac{1}{1 + \omega^k + \omega^{2k} + \omega^{3k}}.\]
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<|im_start|>user
Let $\omega = e^{2 \pi i/1729}.$  Compute
\[\sum_{k = 1}^{1728} \frac{1}{1 + \omega^k + \omega^{2k} + \omega^{3k}}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
432
Let $\omega = e^{2 \pi i/1729}.$  Compute
\[\sum_{k = 1}^{1728} \frac{1}{1 + \omega^k + \omega^{2k} + \omega^{3k}}.\]
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