
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
To commemorate the contribution of the ancient Chinese mathematician Zu Chongzhi to the calculation of pi, the 40th session of the United Nations Educational, Scientific and Cultural Organization (UNESCO) designated March 14th of each year as "International Mathematics Day." On March 14, 2023, a school held a math cultural festival, including a Sudoku competition. Two students, A and B, entered the final round to compete for the title of Sudoku King. The rules of the final round are as follows: there are two rounds of Sudoku competition, each person earns 1 point for correctly solving the puzzle within the specified time, and 0 points for incorrect answers. The Sudoku King is determined by the total score after two rounds, with a tie resulting in a tiebreaker. Based on past performance analysis, it is known that the probability of A solving the puzzle correctly in each round is 0.8, and the probability of B solving the puzzle correctly in each round is 0.75. It is also known that the performance of A and B in each round is independent of each other, and the correctness of their solutions in each round is also independent. 

$(1)$ Find the probability that B scores 1 point after the two rounds of the competition. 

$(2)$ Find the probability that A wins the Sudoku King title without a tiebreaker.

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