
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 basketball league is divided into two stages: preliminary and final. The organizing committee ranks the teams based on their performance in the preliminary stage (assuming the rankings are unique). The top six teams directly advance to the final stage. The seventh and eighth teams play a first revival match, with the winner advancing to the final. The ninth and tenth teams play a match, and the winner plays against the loser of the first revival match in a second revival match, with the winner of this match becoming the last team to enter the final. Assuming that the results of each match do not affect each other and that each match must have a winner.

$(1)$ After the preliminary stage, teams A, B, C, and D rank seventh, eighth, ninth, and tenth, respectively. The probabilities of team D winning against teams A, B, and C are $0.4$, $0.5$, and $0.6$ respectively. The probability of team A winning against team B is $0.6$. What is the probability of team D entering the final?

$(2)$ If teams A and B advance to the final, they need to play a best-of-five series, with the first team to win three games declared the winner. Assuming that the results of each game do not affect each other. If the probability of team B winning each game is $\frac{1}{3}$, and the number of games won by team B when the series ends is denoted by $X$, find the probability distribution table and the mean of $X$.

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