
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 certain shopping mall held a basketball shooting activity to attract customers, with the following rules: if a customer hits three shots in a row, the game is cleared, stopped, and the customer receives a $9$ yuan coupon; if the customer misses two shots in a row, the game fails, stops, and the customer receives a $3$ yuan coupon; if after shooting six times the game is neither cleared nor failed, the game stops, and the customer receives a $6$ yuan coupon. Customer Xiao Ming is ready to participate in this activity, and it is known that Xiao Ming's shooting accuracy rate is $\frac{2}{3}$.  
$(1)$ Find the probability that Xiao Ming ends the game after shooting five times;  
$(2)$ Let $X$ denote the amount of coupons Xiao Ming receives. Find the distribution table of $X$ and its expectation.

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