\[\large \int_0^1 \left\{(-1)^{\left\lfloor 1/x \right\rfloor}\cdot \frac 1x \right\} \ \mathrm{d}x \]

If the value of the above integral can be represented as \(a + \ln \left( \dfrac b\pi \right) \) for positive integers \(a\) and \(b\), find \(a+b\).

**Notations:**

- \(\left\{ \cdot \right\}\) denotes the fractional part function.
- \(\left \lfloor \cdot \right \rfloor \) denotes the floor function.

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