\[f(x) = \begin{cases} \dfrac{\sin (\lfloor x^2 \rfloor \pi)}{x^2-3x-18} +ax^3+b & \text{for } 0 \le x \le 1 \\ 2 \cos (\pi x)+ \tan^{-1} x & \text{for } 1 \le x \le 2 \end{cases} \]

If \(f(x)\) is differentiable in \([0,2]\), find the value of \(\left| \dfrac{\pi}{4} - b - a \right| \).

**Note:**

- \(\lfloor \cdot \rfloor\) denotes the floor function
- \(|\cdot |\) denotes the modulus function.

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