$1.[15$ points$]$ Consider the above expectiminimax tree, where circle nodes are chance nodes, the top node is a min node, and the bottom nodes represent the terminal utilities.

$($a$)[10$ points$]$ Calculate the value of each chance node, as per expectiminimax definition.

$($b$)[5$ points$]$ What is the value of the min node? Which action should the min player take $($e$.$g$.,$ first, second, or third action represented by a chance node$)?$

$($c$)[5$ points$]$ Which nodes can be pruned from the game tree above through alpha$-$beta pruning? Explain the reason if no nodes can be pruned.