
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
On January 15, 2020, the Ministry of Education issued the "Opinions on Carrying out Pilot Reforms in Basic Subject Enrollment in Some Universities" (also known as the "Strong Foundation Plan"). The "Opinions" announced that starting from 2020, universities will no longer organize independent enrollment work, but will implement the Strong Foundation Plan. The Strong Foundation Plan mainly selects and cultivates students who are willing to serve the major strategic needs of the country and have excellent comprehensive qualities or excel in basic subjects. It is known that the written test of the Strong Foundation Plan is independently set by pilot universities, and candidates can only enter the interview stage after passing the written test. It is known that University A and University B both have three exam subjects in the written test, and the passing probability for each subject is independent. If a candidate applies to University A, the probability of passing each subject is $\frac{1}{2}$; if the candidate applies to University B, the passing probabilities for the three subjects are $\frac{1}{6}$, $\frac{3}{5}$, and $m$, where $0 < m < 1$.

$(1)$ If $m = \frac{3}{5}$, find the probability that the candidate applying to University A and University B will pass exactly one subject in the written test.

$(2)$ The Strong Foundation Plan stipulates that each candidate can only apply to one pilot university. If the decision is made based on the mathematical expectation of the number of subjects passed in the written test, and the candidate prefers to pass the written test of University A, find the range of $m$.

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