
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
In the third year of high school, the class organized a fun sports competition. After multiple rounds of competition, Class A and Class B entered the finals. There were three events in the finals, with the winner of each event receiving 2 points and the loser receiving -1 point. There were no draws. The class with the highest total score after the three events would be the champion. It is known that the probabilities of Class A winning in the three events are 0.4, 0.5, and 0.8 respectively, and the results of each event are independent of each other.  
$(1)$ Find the probability of Class A winning the championship.  
$(2)$ Let $X$ represent the total score of Class B. Find the distribution table and expectation of $X$.

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