
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]


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Schools A, B, and C are participating in a basketball competition, each sending a representative team, referred to as Team A, Team B, and Team C. The competition rules are as follows: a team that loses two games in total will be eliminated; before the matches, a draw is held to determine the first two teams to play, while the third team gets a bye; the winning team of each match plays against the team with a bye in the next match, while the losing team gets a bye in the next match, until one team is eliminated; after one team is eliminated, the remaining two teams continue to play until one team is eliminated, and the other team wins the championship, ending the competition. It is known that in each match, the probability of Team A winning against Team B and Team A winning against Team C is both $\frac{2}{3}$, and the probability of Team B winning against Team C is $\frac{1}{2}$. The results of each match are independent. After the draw, Team A gets a bye in the first match.  
$(1)$ Find the probability that "Team B is eliminated after the first three matches";  
$(2)$ Find the probability that "Team A wins the championship in only four matches";  
$(3)$ Find the probability that "a fifth match is needed".

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