
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
There are three manufacturers, A, B, and C, producing products of the same specifications. The qualification rates of the products produced by manufacturers A, B, and C are $0.95$, $0.90$, and $0.80$ respectively. It is known that the proportions of products produced by manufacturers A, B, and C are $2:3:5$.  
$(1)$ Suppose the specifications $Z$ of the products produced by manufacturers A, B, and C follow a normal distribution $N(200,150)$. Find the probability of the event $P(187.8\leqslant Z\leqslant 212.2)$. Given: $\sqrt{150}\approx 12.2$. If $Z\sim N(\mu, \sigma^2)$, then $P(\mu -\sigma \leqslant Z\leqslant \mu +\sigma)\approx 0.6827$, $P(\mu -2\sigma \leqslant Z\leqslant \mu +2\sigma)\approx 0.9545$.  
$(2)$ If the products produced by the three manufacturers are mixed together and one piece is randomly selected from the mixed products, find the probability that this product is qualified.

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