
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
Some universities have carried out pilot reforms in enrolling students in basic disciplines (Strong Foundation Plan), and the school exams are independently set by the pilot universities. In the school exams, candidates who perform well in the written test can enter the interview section. It is known that universities $A$ and $B$ both have three exam subjects in the written test section, and the probability of achieving excellence in each subject is independent of each other. If a candidate applies to university $A$, the probability of achieving excellence in each subject is $\frac{2}{5}$, and if the candidate applies to university $B$, the probabilities of achieving excellence in each subject are $\frac{1}{4}$, $\frac{2}{5}$, and $n$, where $0 \lt n \lt 1$.
$(1)$ If $n=\frac{1}{3}$, find the probabilities that the candidate applying to universities $A$ and $B$ respectively have exactly one subject achieving excellence in the written test section.
$(2)$ According to the Strong Foundation Plan, each candidate can only apply to one pilot university. If the decision is made based on the expectation of the number of subjects in which the candidate achieves excellence in the written test, and the candidate is more likely to enter the interview section of university $A$, find the range of $n$.

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