Let the function \(f\) be twice differentiable on \(\mathbb R\) and also continuous on \(\mathbb R\), and that \(\displaystyle \int_0^\pi (f(x) + f''(x)) \sin x \, dx =\pi \) and \(\displaystyle \lim_{x\to0}\dfrac{f(x)}{\sin x} = 1 \). Find \(f(\pi)\).

\[\] **Notation**: \(\mathbb R \) denotes the set of real numbers.

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