
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 participates in a multi-round fun game, where they draw balls from two opaque boxes, labeled $A$ and $B$. The rules of the game are as follows: in each round, Player A randomly selects $1$ ball from box $A$ and moves it to box $B$, then randomly selects $2$ balls from box $B$. If the results of each round are independent, and before each round, the number of balls in each box remains as shown in the table below (the balls are identical in size and texture except for color):

|        | Red ball | Blue ball | White ball |
|--------|----------|-----------|------------|
| Box $A$ | $2$      | $2$       | $1$        |
| Box $B$ | $2$      | $2$       | $1$        |

$(1)$ Find the probability that in one round, Player A draws white balls from both boxes $A$ and $B$.

$(2)$ Given the scoring rules of the game: if both balls drawn from box $B$ are red, Player A gets $5$ points; if only one red ball is drawn from box $B$, Player A gets $3$ points; if no red balls are drawn from box $B$, Player A gets $1$ point.

$(i)$ Let $X$ denote the score Player A receives in one round. Find the probability distribution of $X$.

$(ii)$ Assuming Player A participates in $5$ rounds of the game, let $Y$ denote the total score obtained by Player A in $5$ rounds. Find $E(Y)$.

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