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A group of friends named \(A_1, A_2, \cdots, A_{2013}\) decide to each play the single-player game AkshajDice. The rules to AkshajDice are as follows

-For \(1 \le k \le 2013\), person \(A_k\) has to roll \(k\) \(2013\)-sided dice numbered \(d_1, d_2, \cdots, d_k\)

-He wins if for all possible pairs of his dice \(d_i\) and \(d_j\), dice \(d_j\) rolls a larger number than dice \(d_i\) if and only if \(j > i\).

Let the expected value of the number of friends who win be \(n\). If \(n\) can be expressed as \(\dfrac{a}{b}\) for relatively prime positive integers \(a,b\), find the sum of the greatest prime factors of \(a\) and \(b\).


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