
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
Yannick picks a number \( N \) randomly from the set of positive integers such that the probability that \( N \) is selected is \( 2^{-N} \) for each positive integer \( N \). He then puts \( N \) identical slips of paper numbered 1 through \( N \) into a hat and gives the hat to Annie. Annie does not know the value of \( N \), but she draws one of the slips uniformly at random and discovers that it is the number 2. What is the expected value of \( N \) given Annie's information?

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