If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor + \dots + \lfloor \frac{2^{2009}}{5} \rfloor$, find the remainder when $2^{2010}$ is divided by $N$.

Let's think step by step and output the final answer within \boxed{}.
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