
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 machine tools, A, B, and C, independently process the same type of parts. The probability that a part processed by machine A is of the first grade and the part processed by machine B is not of the first grade is $\frac{1}{4}$. The probability that a part processed by machine B is of the first grade and the part processed by machine C is not of the first grade is $\frac{1}{12}$. The probability that parts processed by both machines A and C are of the first grade is $\frac{2}{9}$.

1. Determine the probability that the parts processed by machines A, B, and C are of the first grade.
2. If one part is taken from the parts processed by each of machines A, B, and C, find the probability that at least one part is of the first grade.

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