
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
Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:<br/>

|          | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150         | 50           | 200   |
| Machine B | 120         | 80           | 200   |
| Total    | 270         | 130          | 400   |

$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?<br/>
$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?<br/>
Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>

| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$                   | 3.841 | 6.635 | 10.828 |

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