Suppose the polynomial $f(x) = x^{2014}$ is expressed as $f(x) = \sum^{2014}_{k=0} a_k \binom{x}{k}$ for some real numbers $a_0, a_1, \ldots, a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.

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