Sum of Sums

Number Theory Level pending

Many people know the story of Gauss, and how he came up with a method to sum up consecutive integers. However, what many people don’t know is the story of Steelfirez. One day, Steelfirez's particularly cruel teacher (even more so than Gauss') gave him the following problem: Given that \(S_{1}(N) = \sum\limits_{x=1}^{N}x\) and \(S_{k}(N) = \sum\limits_{x=1}^{N}S_{k-1}(x)\), for \(k>1\), what is the closed form of \(S_{k}(N)\)? Steelfirez, having nothing better to do, worked hard and managed to come up with a solution. In honor of Steelfirez's hard work, we ask you to find the maximum positive integer \(p\), such that \(S_{2014}(2014)\equiv0 \ (mod \ 10^p)\).

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