
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
Company $W$'s product $p$ produced by line $D$ follows a normal distribution $N(80, 0.25)$ in terms of size. $400$ randomly selected products were tested from the current production line, and the size distribution is summarized in the table below:

| Product Size ($mm$) | $[76,78.5]$ | $(78.5,79]$ | $(79,79.5]$ | $(79.5,80.5]$ | $(80.5,81]$ | $(81,81.5]$ | $(81.5,83]$ |
|---------------------|-------------|------------|-------------|--------------|------------|-------------|------------|
| Number of Items     | $8$         | $54$       | $54$        | $160$        | $72$       | $40$        | $12$       |

According to the product quality standards and the actual situation of the production line, events outside the range $(\mu - 3\sigma, \mu + 3\sigma]$ are considered rare events. If a rare event occurs, it is considered an abnormality in the production line. Products within $(\mu - 3\sigma, \mu + 3\sigma]$ are considered as qualified products, while those outside are defective. $P(\mu - \sigma < X \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) \approx 0.9545$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) \approx 0.9973$.

$(1)$ Determine if the production line is working normally and explain the reason.

$(2)$ Express the probability in terms of frequency. If $3$ more products are randomly selected from the production line for retesting, with a testing cost of $20$ dollars per qualified product and $30$ dollars per defective product, let the testing cost of these $3$ products be a random variable $X$. Find the expected value and variance of $X.

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