
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
Two groups of students, Team A and Team B, play basketball in their spare time. It is stipulated that the winning team in each game scores $2$ points, while the losing team scores $0$ points. There are no ties. The first team to reach $8$ points wins the game. It is given that the probability of Team A winning a game is $\frac{2}{3}$.  
$(1)$ Find the probability that the game ends after exactly $6$ rounds.  
$(2)$ If Team A is leading with a score of $4:2$ currently, let $X$ represent the number of rounds needed to end the game. Find the probability distribution and the expected value of $X$.

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