
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 factory produces a batch of components, with a diameter $X$ following a normal distribution $X\sim N\left(10,0.25\right)$ (unit: $mm)$<br/>$(1)$ Now, $15$ components are randomly selected for testing. Products with diameters within $\left(8.5,11.5\right)$ are considered qualified. If defective products are found in the sample, it can be concluded that there is a problem in the production process. Find the probability of the above event occurring and explain the rationality of this standard. (Given: $P\left(\mu -3\sigma\ \ \lt X \lt \mu +3\sigma \right)\approx 0.9973$, $0.9973^{15}\approx 0.9603)$<br/>$(2)$ If an issue is found in the above test, another $100$ components are randomly selected for further testing. What is the most likely number of defective components among these $100$ components?

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