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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