Let \(S\) be the set of numbers less than \(5\) billions and can be expressed in the form \(n^2+5n+23\), where \(n\) is a positive integer.

Let \(N\) be the smallest prime number that divides for some elements of \(S\).

Let \(p\) be the number of elements of \(S\) that are prime.

Let \(q\) be the number of elements of \(S\) that are divisible by \(N\) .

Find \(N+p+q\).

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