
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
Suppose there is a bag containing four balls of different colors: red, yellow, blue, and green. Each time a ball is drawn from the bag, its color is recorded and the ball is returned to the bag. This process continues until two consecutive red balls are drawn. Let $\zeta$ be the total number of draws. If the probability of drawing a ball of any color is the same each time, what is the expected value of $\zeta$?

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