
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
The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excellent comprehensive quality or outstanding basic subject knowledge. The school exam of the Strong Foundation Plan is independently formulated by pilot universities. In the school exam process of a certain pilot university, students can only enter the interview stage after passing the written test. The written test score $X$ of students applying to this pilot university in 2022 approximately follows a normal distribution $N(\mu, \sigma^2)$. Here, $\mu$ is approximated by the sample mean, and $\sigma^2$ is approximated by the sample variance $s^2$. It is known that the approximate value of $\mu$ is $76.5$ and the approximate value of $s$ is $5.5$. Consider the sample to estimate the population.

$(1)$ Assuming that $84.135\%$ of students scored higher than the expected average score of the university, what is the approximate expected average score of the university?

$(2)$ If the written test score is above $76.5$ points to enter the interview, and $10$ students are randomly selected from those who applied to this pilot university, with the number of students entering the interview denoted as $\xi$, find the expected value of the random variable $\xi$.

$(3)$ Four students, named A, B, C, and D, have entered the interview, and their probabilities of passing the interview are $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{2}$, and $\frac{1}{2}$, respectively. Let $X$ be the number of students among these four who pass the interview. Find the probability distribution and the mathematical expectation of the random variable $X$.

Reference data: If $X \sim N(\mu, \sigma^2)$, then: $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$.

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