
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
A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box each time. Winning is defined as drawing two balls of the same color, while losing is defined as drawing two balls of different colors.  
$(1)$ If it is specified that after the first draw, the balls are put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $X$.  
$(2)$ If it is specified that after the first draw, the balls are not put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $Y$.

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