Somewhat Big Numbers

A sequence of pairs of integers \((x_0,y_0),\ (x_1,y_1),\ (x_2,y_2),...\) is defined, starting from some initial pair \((x_0,y_0),\) by the following formulas: For all positive integers \(k\), \[\begin{cases} x_{k}=x_{k-1}^2y_{k-1}\\ y_{k}=x_{k-1}^2y_{k-1}+x_{k-1}y_{k-1}^2. \end{cases}\] Find the smallest \(N\) such that \(x_N\) is divisible by \(100!\) for every initial pair of starting values \((x_0,y_0).\)

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