Sum 1 follow-up

Calculus Level pending

This problem is a follow-up to "Sum 1", my previous problem; if you've read my solution to "Sum 1", this problem should be straightforward.

Evaluate:

\[\sum_{n=1}^{\infty} {\left(\frac{\left(-1\right)^n}{n}\sum_{k=1}^n {\left(\frac{\left(-1\right)^k}{k}\right)}\right)}\]

The answer can be expressed as \(\dfrac{\pi^{a_1}}{6a_2}+\dfrac{\left(\ln{a_3}\right)^{a_4}}{a_5}\), where \(a_1\), \(a_2\), \(a_3\), \(a_4\), and \(a_5\) are prime positive integers; find \(10000a_1+1000a_2+100a_3+10a_4+a_5\).

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