
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
A company organized a lottery event for its employees to win shopping coupons. The rules are as follows: each person randomly draws 3 balls at once from a box containing 4 yellow balls and 4 red balls of uniform size and texture. If exactly 1 red ball is drawn, the person wins a $50 coupon; if exactly 2 red balls are drawn, the person wins a $100 coupon; if all 3 balls are red, the person wins a $200 coupon. Otherwise, no coupon is awarded. Xiao Wang participated in the company's lottery event.  
$(1)$ Find the probability that Xiao Wang draws exactly 1 yellow ball.  
$(2)$ Let X be the amount of coupon Xiao Wang wins. Find the distribution table and expectation of X.

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