# Complicated Infinite Sum

$\large \displaystyle\sum_{n = 1}^{\infty}\frac{d(n) + \displaystyle\sum_{m = 1}^{v_2(n)}(m - 3)d\left(\frac{n}{2^m}\right)}{n}$

Let $$S$$ be the value of the summation above, where $$d(n)$$ is the number of divisors of $$n$$ and $$v_2(n)$$ be the exponent of $$2$$ in the prime factorization of $$n$$. If $$S = (\ln m)^n$$ for positive integers $$m$$ and $$n$$, find $$1000n + m$$.

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