
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

...

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Suppose students A and B go to school every day, and the probability of each of them arriving before 7:30 is $\frac{2}{3}$. Assume that the arrival of students A and B are independent events, and the daily arrival of each student is also independent.
(I) Using X to represent the number of days student A arrives before 7:30 during three school days, find the distribution list and mathematical expectation of the random variable X.
(II) Let M be the event that "During three school days, the number of days student A arrives before 7:30 is exactly 2 more than the number of days student B arrives before 7:30." Find the probability of event M occurring.

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