For every positive integer $n$, define $S_n$ to be the sum
\[
  S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \pi}{2010} \right)^n .
\]
As $n$ approaches infinity, what value does $S_n$ approach?

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