
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
In the movie "The Wandering Earth 2," there are many UEG (United Earth Government) mechanical devices that are drivable, operable, and deformable, all of which are from the leading Chinese engineering machinery brand - XCMG. Many of the hardcore equipment in the movie are not special effects, but are actually designed and modified from cutting-edge domestic equipment. Many of the equipment can find prototypes in reality. A new device has been developed in a workshop of the group. The specific requirement of the group for the new device is: the product with a part inner diameter (unit: mm) within the range of $(199.82, 200.18)$ is qualified, otherwise it is defective; the part inner diameter $X$ follows a normal distribution $X \sim N(200, 0.0036)$.

$(1)$ If the workshop installed and debugged the new device and produced 5 parts for trial, with measured inner diameters (unit: mm) of $199.87$, $199.91$, $199.99$, $200.13$, $200.19$, if you are in charge of the workshop, try to determine whether this device needs further debugging based on the $3\sigma$ principle? Explain your reasoning.

$(2)$ If the device meets the production requirements of the group, and now 10,000 parts produced by this device are tracked.

① Approximately how many parts out of the 10,000 parts have an inner diameter exceeding $200.12$ mm?

② What is the most likely number of defective parts out of the 10,000 parts?

Reference data:

If a random variable $X \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma < X < \mu + \sigma) \approx 0.683$, $P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.955$, $P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 0.997$, $0.997^4 \approx 0.988$, $0.997^5 \approx 0.985$.

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