
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
Assume there is a bag with red, yellow, blue, and green balls. Each time, one ball is taken from the bag, the color is confirmed, and then the ball is put back into the bag. This process continues until two consecutive red balls are drawn. Let $\zeta$ be the total number of draws. If each ball is equally likely to be drawn, what is the expected value of $\zeta$?

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