
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 promote the national "livelihood project," a certain city government is now providing a batch of affordable housing to ensure residents have housing. There are four sets of identical conditions available for applicants A, B, C, and D, with 3 people applying for them, and their applications are independent of each other.

(1) Find the probability that applicants A and B do not apply for the same set of housing.

(2) Let X be the number of people among the 3 applicants who apply for set A. Find the distribution table and mathematical expectation of X.

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