
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Two people, A and B, take turns shooting basketball. The rules are as follows: if a player makes a shot, they continue shooting; if they miss, the other player takes the next shot. Regardless of previous shots, player A has a shooting percentage of $0.6$ and player B has a shooting percentage of $0.8$. The person to take the first shot is determined by drawing lots, with the probabilities of A and B being selected for the first shot both being $0.5$.

$(1)$ Find the probability that player B takes the second shot.

$(2)$ Find the probability that player A takes the $i$-th shot.

$(3)$ Let the random event $Y$ represent the number of shots player A takes, $Y=0,1,2,\ldots,n$. Find $E(Y)$.

Present the answer in LaTex format: \boxed{Your answer}
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