
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
Player A and player B take turns shooting baskets. 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 outcomes, player A has a shooting percentage of $0.6$ and player B has a shooting percentage of $0.8$. The player for the first shot is determined by drawing lots, with player A and player B each having a probability of $0.5$.  
$(1)$ Find the probability that player B takes the second shot;  
$(2)$ Find the probability that player A takes the $i$-th shot;  
$(3)$ Given: if the random variable $X_{i}$ follows a two-point distribution, and $P(X_{i}=1)=1-P(X_{i}=0)=q_{i}$, $i=1, 2, \cdots, n$, then $E(\sum_{i=1}^n{{X_i}})=\sum_{i=1}^n{{q_i}}$. Let the number of times player A shoots in the first $n$ shots be denoted as $Y$, find $E\left(Y\right)$.

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