$(12$ points$)$ Consider a binary symmetric channel with memory. For the $n$th channel use, the binary output $Y[n]in\{0,1\}$ is related to the binary input $x[n]in\{0,1\}$ by

$Y[n]=x[n]o+Z[n]$

where $Z[n]in\{0,1\}$ is a binary random variable with $P\{Z[n]=0\}=p,$ and $P\{Z[n]=1\}=1-p.$ Assume that for every integer $k,Z[2k]=Z[2k+1],$ i$.$e$.,$ the noises of two successive channel uses are exactly the same. Obtain the capacity of this channel in bit per channel use.

$($hint: Consider two successive channel uses together to construct a new vector channel model.$)$