Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\prod_{k=1}^{n}p_k^{a_k}$, where the $p_k$'s are distinct primes and the $a_k$'s are positive integers. Find $\sum_{k=1}^{n}(p_k + a_k)$.

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