Consider the signals $x\left [ n \right ]=2^{n-1}u\left [ -n+2 \right ]$ and $y\left [ n \right ]=2^{-n+2}u\left [ n+1 \right ]$, where $u[n]$ is the unit step sequence. Let $X(e^{jw})$ and $Y(e^{jw})$ be the discrete-time Fourier transform of $x[n]$ and $y[n]$, respectively. The value of the integral
$$\frac{1}{2\pi } \int_{0}^{2\pi }X\left ( e^{j\omega } \right )Y\left ( e^{-j\omega } \right )d\omega$$
(rounded off to one decimal place) is ___________