
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
The world is currently undergoing a major transformation that has not been seen in a century. China is facing new challenges. In order to enhance students' patriotism and cohesion, a certain high school organized a knowledge competition on "China's national conditions and the current world situation" for the second year students. The main purpose is to deepen the understanding of the achievements China has made in economic construction, technological innovation, and spiritual civilization construction since the founding of the People's Republic of China, as well as the latest world economic and political current affairs. The organizers randomly divided the participants into several groups by class. Each group consists of two players. At the beginning of each match, the organizers randomly select 2 questions from the prepared questions for the two players to answer. Each player has an equal chance to answer each question. The scoring rules are as follows: if a player answers a question correctly, they get 10 points, and the other player gets 0 points; if a player answers a question incorrectly or does not answer, they get 0 points, and the other player gets 5 points. The player with more points after the 2 questions wins. It is known that two players, A and B, are placed in the same group for the match. The probability that player A answers a question correctly is 2/3, and the probability that player B answers a question correctly is 4/5. The correctness of each player's answer to each question is independent. The scores obtained after answering the 2 questions are the individual total scores of the two players.
  
$(1)$ Find the probability that player B's total score is 10 points;  
$(2)$ Let X be the total score of player A. Find the distribution and mathematical expectation of X.

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