We are given 5040 balls in \(k\) different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest \(k\) such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no two balls of the same color are in two neighboring bags.

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