
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 promote the spirit of sports and create a campus sports atmosphere, a certain school organized a "Youth Cup" 3v3 basketball competition, with teams A and B entering the finals. The rules are as follows: the team that accumulates two wins first will be the champion. A player who commits more than 4 fouls in a game will not be able to participate in the following games after that game. Under the allowed rules, player M from team A will participate in all games. The probabilities of team A winning a game when player M is on the court and when he is not on the court are $\frac{3}{5}$ and $\frac{2}{5}$, respectively. The probability of committing more than 4 fouls in a game is $\frac{1}{4}$.

$(1)$ Find the probability of team A winning the second game.

$(2)$ Let $X$ represent the number of games when the competition ends, find the expected value of $X$.

$(3)$ Given that player M committed more than 4 fouls in the first game, find the probability of team A winning the game.

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