$5\mathrm{.}20.$ MAC with a helper. Consider the $3-$sender DM$-$MAC p$($y$|$x$\_(1),$x$\_(2),$x$\_(3))$ with mes$-$ sage pair $($M$\_(1),$M$\_(2))$ uniformly distributed over $[1$:$2^($nR$\_(1))]\backslash$times $[1$:$2^($nR$\_(2))]$ as depicted in Figure $5\mathrm{.}12.$ Sender $1$ encodes M$\_(1),$ sender $2$ encodes M$\_(2),$ and sender $3$ encodes $($M$\_(1),$M$\_(2)).$ The receiver wishes to recover both messages. Show that the capacity region is the set of rate pairs $($R$\_(1),$R$\_(2))$ such that R$\_(1)<=$I$($x$\_(1),$x$\_(3)$;Y$|$x$\_(2),$Q$),$ R$\_(2)<=$I$($x$\_(2),$x$\_(3)$;Y$|$x$\_(1),$Q$),$ R$\_(1)+$R$\_(2)<=$I$($x$\_(1),$x$\_(2),$x$\_(3)$;Y$|$Q$)$ for some pmfp$($q$)$p$($x$\_(1)|$q$)$p$($x$\_(2)|$q$)$p$($x$\_(3)|$x$\_(1),$x$\_(2),$q$)$ with $|$Q$|<=2.$ Remark: This is a special case of the capacity region of the general l $-$message k $-$ sender DM$-$MAC established by Han $(1979).$ Figure $5\mathrm{.}12.$ Three$-$sender DM$-$MAC with two messages.