Two bags of balls each contain balls numbered \(1, 2, \ldots , n\). For how many values of \(n\) such that \(1 \leq n \leq 500\), does there exist a \(k\) such that the probability of getting a sum of \(n+1\) when a ball is selected from each bag is equal to the probability that the sum is \(\leq k\) when a ball is selected from each bag?

**Details and assumptions**

Since a ball is selected from each bag, there are 2 balls selected, and we are looking at the sum of these values.

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