
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 certain middle school conducted a patriotic knowledge competition for the entire school, with a total of 1000 students participating. The distribution of the number of correct answers (out of 60 questions) is shown in the table below:

| Number of Correct Answers | $[0,10)$ | $[10,20)$ | $[20,30)$ | $[30,40)$ | $[40,50)$ | $[50,60]$ |
|---------------------------|----------|-----------|-----------|-----------|-----------|-----------|
| Frequency                 | $10$     | $185$     | $265$     | $400$     | $115$     | $25$      |

The number of correct answers $Y$ approximately follows a normal distribution $N(\mu, 81)$, where $\mu$ is the average number of correct answers for these 1000 people (using the midpoint value of the interval as the representative value).

$(1)$ Estimate the number of people with correct answers in the range $[12,48]$ (rounded to the nearest integer).

$(2)$ Considering the frequency as probability, now randomly select 4 students from this school. Let $X$ be the number of students with correct answers in the range $[30,40)$. Find the probability mass function and the expected value of $X$.

Given: If $Z \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma \leq Z \leq \mu + \sigma) = 68.3\%$, $P(\mu - 2\sigma \leq Z \leq \mu + 2\sigma) = 95.4\%$, $P(\mu - 3\sigma \leq Z \leq \mu + 3\sigma) = 99.7\%$.

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