
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
During a promotion in a certain mall, customers who spend $500$ yuan or more are eligible to participate in a lucky draw. The rules are as follows: each customer spins a uniform disk labeled with $1$, $2$, $3$, $4$, $5$, $6$ three times. If the pointer points to "$4$" once or twice, the customer will receive a refund of $30\%$ of the purchase amount from the mall; if the pointer points to "$4$" three times, the customer will receive a refund of $50\% of the purchase amount; otherwise, no refund will be given. A customer's total purchase amount on the receipt during this promotion period is $800$ yuan.  
$(1)$ Find the probability that the customer will receive exactly a $240$ yuan refund after participating in the activity;  
$(2)$ Suppose that after participating in the activity, the final amount the customer pays to the mall is $Y$, find the distribution and mathematical expectation of $Y$. (Round to the nearest integer)

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