
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 type of alloy steel production company needs to ensure that the carbon content percentage of the alloy steel is within a specified range. Under the same test conditions, the inspector randomly samples $10$ times each day and measures the carbon content (unit: $\%$). It is known that the carbon content of their product follows a normal distribution $N(\mu, \sigma^2)$.

$(1)$ Assuming the production is normal, let $X$ represent the number of times in a day of $10$ samples where the carbon content percentage is outside of $(\mu - 3\sigma, \mu + 3\sigma)$. Find $P(X \geq 1)$ and the expected value of $X$.

$(2)$ In a day of inspections, if there is at least one detection of carbon content outside of $(\mu - 3\sigma, \mu + 3\sigma)$, it is considered that there may be an abnormality in the production process that day, and the production process of that day needs to be inspected. The following are the measurement results of carbon content (unit: $\%$) obtained by the inspector in $10$ tests in a day:

| Number | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|--------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|
| Carbon Content $\left(\%\right)$ | $0.31$ | $0.32$ | $0.34$ | $0.31$ | $0.30$ | $0.31$ | $0.32$ | $0.31$ | $0.33$ | $0.32$ |

After calculation, it is found that $\overline{x} = \frac{1}{10}\sum_{i=1}^{10}x^{i} = 0.317$ and $s = \sqrt{\frac{1}{10}\sum_{i=1}^{10}({x}_{i}-\overline{x})^{2}} = 0.011$, where $x_{i}$ is the carbon content percentage of the $i$-th sample $(i=1,2,\cdots,10)$.

$(i)$ Using the sample mean $\overline{x}$ as the estimate $\hat{\mu}$ of $\mu$ and the sample standard deviation $s$ as the estimate $\hat{\sigma}$ of $\sigma$, determine if it is necessary to inspect the production process of that day.

$(ii)$ If $x_{1}$ is removed, and the mean and standard deviation of the remaining data are denoted as $\mu_{1}$ and $\sigma_{1}$, respectively, write the formula for $\sigma_{1}$ in terms of $\overline{x}$, $s$, $x_{1}$, and $\mu_{1}$. 

Note: If a random variable $Z$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - 3\sigma < Z < \mu + 3\sigma) = 0.9974$. $0.9974^{10} \approx 0.9743$.

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