Power Trap

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: \[\begin{cases} x_{k+1}=x_k^4y_k-x_k^3y_k^2\\ y_{k+1}=x_k^2y_k^3-x_ky_k^4 \end{cases}\] Find the number of primes \(p<1000\) such that \(x_{10}\) is a multiple of \(p\) for all initial pairs \((x_0,y_0)\).

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