
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
An organization holds a knowledge competition with two types of questions, type A and type B. Three employees, $A$, $B$, and $C$, participate in the competition. The rules of the competition are as follows: a question is randomly selected from type A questions for the employee to answer. If the answer is incorrect, the competition for that employee ends. If the answer is correct, a question is then randomly selected from type B questions for the employee to answer. Regardless of whether the answer is correct or not, the competition for that employee ends. The process of answering questions twice is independent for each person, and the three people answering questions are also independent. In type A questions, each correct answer earns 20 points, otherwise, 0 points are earned; in type B questions, each correct answer earns 80 points, otherwise, 0 points are earned. It is known that the probability of employee $A$ correctly answering type A questions is 0.5 and correctly answering type B questions is 0.6; the probability of employee $B$ correctly answering type A questions is 0.6 and correctly answering type B questions is 0.5; the probability of employee $C$ correctly answering type A questions is 0.4 and correctly answering type B questions is 0.75.  
$(1)$ Find the probability that the sum of the scores of the three people is 20 points;  
$(2)$ Let the random variable $X$ be the number of people among the three who score 100 points, find the expected value of the random variable $X$.

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