
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
To enrich students' extracurricular activities, the school badminton club held a badminton team competition. The competition adopted a format of 5 games with the first team to win 3 games being declared the winner. Each game is played in singles mode, with each team consisting of 5 members. Each player can only play once during the match, and the order of play is random. The results of each game do not affect the others. After the group stage, teams A and B advanced to the final. Based on the statistics from the previous matches, the star player $M$ from team A has a winning rate of $\frac{3}{4}$ against each player from team B, while the other 4 members of team A have a winning rate of $\frac{1}{2}$ against each player from team B. (Note: There are no draws in the matches)

$(1)$ If the star player $M$ from team A plays in the first three games, let $X$ denote the number of games team A wins in the first three games. Find the probability distribution and the expected value of the random variable $X$.

$(2)$ If it is known that teams A and B played 3 games, and team A won with a score of $3:0$, find the probability that the star player $M$ from team A played in the match.

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