
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
Three students, A, B, and C, are playing badminton with the following rules:<br/>The player who loses two games in a row will be eliminated. Before the game, two players are randomly selected to play against each other, while the third player gets a bye. The winner of each game plays against the player with the bye in the next round, and the loser gets the bye. This process continues until one player is eliminated. After one player is eliminated, the remaining two players continue playing until one of them is eliminated, and the other player wins, ending the game. After drawing lots, A and B play first, while C gets the bye. The probability of each player winning a game is $\frac{1}{2}$.<br/>$(1)$ Find the probability of A winning four consecutive games;<br/>$(2)$ Find the probability of needing a fifth game to be played;<br/>$(3)$ Find the probability of C winning in the end.

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