
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
Two students, A and B, are playing chess. It is known that when A goes first, the probability of A winning is $\frac{2}{3}$, the probability of a draw is $\frac{1}{6}$, when B goes first, the probability of B winning is $\frac{1}{2}$, and the probability of a draw is $\frac{1}{4}$. In the first round, A goes first, and the order of who goes first in the following matches is as follows: if there is a winner in the previous round, the loser of the previous round goes first in the next round; if the previous round was a draw, then B goes first in the next round. The results between each match are independent of each other. If a player wins three rounds first, that player wins and the game ends.  
$(1)$ Find the probability of the game ending within three rounds and A winning all three rounds;  
$(2)$ Find the probability of the game ending within five rounds and B winning.

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