How many Transformations?

Algebra Level 5

Let \(N\) be a positive integer that satisfies

\[\left(\begin{array}{cc}2 & 0 \\ 1 & 4\end{array}\right)^N\cdot \dbinom{2}{1}=\dbinom{x}{y}.\]

Find the largest possible value of \(N\) such that

\[\log_2(xy) \le 1000.\]

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