Khang, Calvin, and Brian play the following game. Given a fixed positive integer \(k\) which is at most 2016, they randomly choose a subset A of \(\{1, 2, \ldots, 2016\}\) with \(k\) elements.

The winner is Khang, Calvin, or Brian, respectively, if the sum of the numbers in \(A\) leaves a remainder of 0, 1, or 2 when divided by 3.

How many values of \(k\) for which this game is fair (i.e., such that the three possible outcomes are equally likely)?

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