
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
Two sports schools, School A and School B, each have three subjects for examination: soccer, long-distance running, and long jump. If Xiaoming applies to School A, the probability of passing each subject is $\frac{2}{3}$. If Xiaoming applies to School B, the probabilities of passing soccer, long-distance running, and long jump are $\frac{3}{4}$, $\frac{2}{3}$, and $m$ respectively, where $0 < m < 1$, and the passing of each subject is independent of each other.  
$(Ⅰ)$ If $m = \frac{3}{4}$, let $A$ represent the event "Xiaoming passes exactly $2$ subjects when applying to School A," and $B$ represent the event "Xiaoming passes at most $2$ subjects when applying to School B." Find $P(A)$ and $P(B)$.  
$(Ⅱ)$ If Xiaoming's expected number of passed subjects is greater when applying to School A compared to School B, find the range of values for $m$.

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