The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. Find how many pairs of integers $(m,n)$ there are such that $1\leq m\leq 2012$ and  
$$5^n<2^m<2^{m+2}<5^{n+1}.$$

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