\[ \cfrac{i\left(\operatorname{Li}_2\left(\cfrac{i+1}{2}\right)-\operatorname{Li}_2\left(i\right)+\operatorname{Li}_2\left(-i\right)-\operatorname{Li}_2\left(-\cfrac{i-1}{2}\right)\right)}{2} \]

If the above can be expressed in the form of \(\cfrac{\pi \ln{(q)}}{r}\), where \(q\) and \(r\) are both positive integers and \(q\) being a prime, find \( q + r \).

**Notations:**

- \(i = \sqrt{-1}\) is the imaginary unit.
- \( \operatorname{Li}_s (z) \) denotes the polylogarithm function.

For more problems like this, try answering this set .

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