
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
Three machines, labeled A, B, and C, independently manufacture the same type of component. The probability that machine A produces a first-class component while machine B does not is $\frac{1}{4}$. The probability that machine B produces a first-class component while machine C does not is $\frac{1}{12}$. The probability that both machines A and C produce first-class components is $\frac{2}{9}$.

1. Calculate the probability that each machine produces a first-class component.
2. If one component is randomly selected from each machine for inspection, calculate the probability that at least one of them is a first-class component.

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