# 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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