
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
$A$ and $B$ are two drugs used to treat the same disease. A comparative experiment is conducted with several test groups, each consisting of 4 mice, with 2 mice taking drug $A$ and the other 2 taking drug $B$. The effectiveness is observed, and if the number of mice effectively treated with drug $A$ in a test group is greater than those treated with drug $B$, the test group is called a Type A group. It is given that the probability of a mouse being effectively treated with drug $A$ is $\frac{2}{3}$, and with drug $B$ is $\frac{1}{2}$.

$(1)$ Find the probability of a test group being a Type A group.  
$(2)$ Observing 3 test groups, find the probability that at least one of them is a Type A group.

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