AMC 12 2014a Problem 22

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $$5^n<2^m<2^{m+2}<5^{n+1}?$$ $\textbf{(A) }278 \textbf{(B) }279 \textbf{(C) }280 \textbf{(D) }281 \textbf{(E) }282$