Somewhat Big Numbers

A sequence of pairs of integers (x0,y0), (x1,y1), (x2,y2),...(x_0,y_0),\ (x_1,y_1),\ (x_2,y_2),... is defined, starting from some initial pair (x0,y0),(x_0,y_0), by the following formulas: For all positive integers kk, {xk=xk12yk1yk=xk12yk1+xk1yk12.\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 NN such that xNx_N is divisible by 100!100! for every initial pair of starting values (x0,y0).(x_0,y_0).

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