
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
$M_{1}$ and $M_{2}$ are two new drugs for treating the same disease. A research and development company conducts comparative experiments with several test groups. Each test group consists of 4 mice, with 2 mice taking $M_{1}$ and the other 2 taking $M_{2}$, and then observes the efficacy. If in a test group, the number of mice effectively treated with $M_{1}$ is more than those treated with $M_{2}$, the test group is called a superior group. It is given that each mouse has a probability of $\frac{1}{2}$ of being effectively treated with $M_{1}$ and a probability of $\frac{1}{3}$ of being effectively treated with $M_{2}$.
$(Ⅰ)$ Find the probability that a test group is a superior group.
$(Ⅱ)$ Observe 3 test groups, let $\xi$ represent the number of superior groups among these 3 test groups, and find the probability distribution and mathematical expectation of $\xi$.

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