$(1)$ A game is played on the board below. You start at the middle node. On each turn,

you spin a spinner to choose a random color $($with equal probabilities$)$ and then

move along the arrow of that color to the next node. When you reach one of the

three colored nodes, the game ends and you get the number of points displayed

inside the node.

$($a$)$ Determine, for each of the four blank nodes, the expected number of points you

will eventually get if you are currently at that node.

$($b$)$ Starting from the middle node, what are the probabilities that you will end up

at the red node, blue node, and green node?

$($c$)$ Starting from the middle node, on average how many moves will it take before

the game is over?