Consider the curve \( C_{1} \) given by \( y=e^{-x} \) for \( x\in[0,10\pi] \), and the curve \( C_{2} \) given by \( y=e^{-x}(\sin x+\cos x) \) for \( x\in[0,10\pi] \).
Let \( n \) be the total number of points of intersection of the curves \( C_{1} \) and \( C_{2} \).
Suppose that \( \alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in[0,10\pi] \) are the \( x \)-coordinates of the points of intersection of the curves \( C_{1} \) and \( C_{2} \) such that \( \alpha_{1}<\alpha_{2}<\ldots<\alpha_{n} \).
Let \( \beta \) be the area of the region enclosed between the curves \( C_{1}, C_{2} \), and the lines \( x=\alpha_{1} \) and \( x=\alpha_{4} \). Then the value of \( -\frac{1}{\pi}\log_{e}\!\left(\beta-2e^{-\pi/2}\right) \) is __________.
