Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = e^{-x} \sin x \). If \( F : [0,1] \to \mathbb{R} \) is a differentiable function such that \( F(x) = \displaystyle\int\limits_{0}^{x} f(t)\, dt \), the value of \( \displaystyle \int\limits_{0}^{1} \left(F^{\prime}(x) + f(x)\right)e^{x} dx \) lies in the interval.
