Please provide solution for this question.k $-$receiver Gaussian BC$.$ Consider the k $-$receiver Gaussian BC Y$\_(1)=$x$+$Z$\_(1),$ Y$\_($j$)=$Y$\_($j$-1)+$tilde$($Z$)\_($j$),$jin$[2$:k$],$ where Z$\_(1),$tilde$($Z$)\_(2),$dots,tilde$($Z$)\_($k$)$ are independent Gaussian noise components with powers N$\_(1),$tilde$($N$)\_(2),$dots,tilde$($N$)\_($k$),$ respectively. Assume average power constraint P on x $.$ Provide a characterization of the private$-$message capacity region in terms of the noise pow$-$ ers, P $,$ and power allocation parameters $\backslash$alpha $\_(1),$dots,$\backslash$alpha $\_($k$)>=0$ with $\_($j$=1)^$k $\backslash$alpha $\_($j$)=1.$