
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
Jar A originally contains 6 green buttons, 3 red buttons, and 9 blue buttons. Carla moves a number of green and blue buttons to an empty Jar B such that the number of blue buttons moved is twice the number of green buttons moved. After this transfer, Jar A contains only half of its original number of buttons. What is the probability that a randomly chosen button from Jar A is blue, and a randomly chosen button from Jar B is a green button? Express your answer as a common fraction.

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