
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
In a shopping lottery event, suppose there is one first prize coupon worth $50$ yuan among $10$ coupons, three second prize coupons worth $10$ yuan each, and the remaining six coupons have no prize. A customer draws $2$ coupons from these $10$ coupons. Find:<br/>$(1)$ The probability that the customer wins a prize.<br/>$(2)$ The probability distribution table and the mathematical expectation of the total value $\xi$ of the prizes the customer wins.

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