Consider the curve \( C_1 \) given by
\( y = e^{-x} \quad \text{for } x \in [0, 10\pi], \)
and the curve \( C_2 \) given by
\(y = e^{-x}(\sin x + \cos x) \quad \text{for x \in [0, 10\pi]. \)
Let \( n \) be the total number of points of intersection of the curves \( C_1 \) and \( C_2 \).
Suppose that \( \alpha_1, \alpha_2, \dots, \alpha_n \in [0, 10\pi] \) are the \( x \)-coordinates of the points of intersection of the curves \( C_1 \) and \( C_2 \) such that
\(\alpha_1 < \alpha_2 < \dots < \alpha_n. \)}
15.
Then the value of $n$ is {