
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 students, A and B, play badminton in their spare time. In each game, the winner earns 2 points, the loser earns 0 points, and there are no draws. The first player to reach 10 points wins the game. It is assumed that the probability of player A winning each game is $\frac{1}{3}$.  
$(1)$ Find the probability that exactly 6 games are played before the game ends.  
$(2)$ If player A is leading with a score of 6:2, let $X$ represent the number of games left to finish 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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