
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
Twelve balls numbered from $1$ to $12$ are in a jar. Jack reaches into the jar and removes one ball randomly, and then Jill reaches into the jar and randomly removes another ball, different from the one Jack removed. The probability that the sum of the numbers on the balls is even and at least one of the balls is a prime number is required.
A) $\frac{4}{9}$
B) $\frac{9}{19}$
C) $\frac{1}{2}$
D) $\frac{10}{19}$
E) $\frac{11}{132}$

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