
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 balls, one red and one green, are randomly and independently tossed into bins numbered with the positive integers such that the probability that either ball is tossed into bin $k$ is $\frac{1}{2^k}$, for $k = 1, 2, 3, \dots$. Calculate the probability that both balls land in even-numbered bins.

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