$[$ X def $=$ b$.$c$\mathrm{.}0+$ b$.$d$\mathrm{.}0$ Z def $=$ a $$ Y$+$a $$ X; Y def $=$ X$+$b $($c $\mathrm{.}0+$ d$\mathrm{.}0)$ W def $=$ a $$ Y $]$

$($a$)$ Draw a labelled transition system which includes the above states X$,$ Y$,$ Z and W$.[5$ marks$]($b$)$ Explain why Z $2$ but Z $\_3$ W$.$ That is$,$ argue that the second player has the winning strategy in the copy$-$cat $($bisimulation$)$ game which lasts for two rounds, but that the first player has the winning strategy in the game which lasts for three rounds. $[3$ marks$]($c$)$ Give a formula P of the modal logic M which is satisfied by Z but not by W$.[2$ marks$]$