
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
Player A and player B are preparing for a badminton match. The rules state that the winner of a round will serve in the next round. If player A serves, the probability of player A winning the round is $\frac{3}{4}$; if player B serves, the probability of player A winning the round is $\frac{1}{4}$. The results of each round are independent. By drawing lots, it is determined that player A will serve in the first round.  
$(1)$ Find the probability that player B will serve in the third round.  
$(2)$ Find the probability that the number of rounds won by player A in the first three rounds is not less than the number won by player B.

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