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 \(1000n + m\).

Source: OMO
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