
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
To accelerate the construction of a strong manufacturing city in Shaoxing, the "Shaoxing Implementation Plan of Made in China 2025" points out that by the year 2025, the key areas of the manufacturing industry will comprehensively achieve intelligence, and the transformation and upgrading from "Made in Shaoxing" to "Smart Manufacturing in Shaoxing" will be basically realized. A pilot enterprise upgrades its existing production equipment for monitoring the effect of the transformation. Recently, $10$ products are randomly sampled from the production line every day, and a certain quality indicator is analyzed. Based on long-term experience, it can be considered that the quality indicator of products produced under normal conditions of the new equipment follows a normal distribution $N(\mu, \sigma^2)$.

$(1)$ Let $X$ represent the number of products with quality indicators outside of $(\mu - 3\sigma, \mu + 3\sigma)$ among the $10$ products sampled in a day. Find $P(X \geq 1)$.

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

$(2)$ The quality indicators of $10$ products sampled in a day are as follows:

| $9.85$ | $10.12$ | $10.02$ | $9.89$ | $10.21$ |
|--------|---------|---------|--------|---------|
| $10.26$| $9.91$ | $10.13$ | $10.17$| $9.94$  |

If a quality indicator greater than $10.10$ is considered as a first-class product, now $4$ products are randomly selected from the above $10$ products. Let $Y$ be the number of first-class products among these $4$ products. Find the distribution table and the mathematical expectation of $Y$.

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