
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

Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on \( [0,1] \). To generate this number, Chebby uses the factorial base: the number \( 0 . A_{1} A_{2} A_{3} A_{4} \ldots \) stands for \(\sum_{n=0}^{\infty} \frac{A_{n}}{(n+1)!}\), where each \( A_{n} \) is an integer between 0 and \( n \), inclusive. Chebby has an infinite number of cards labeled 0, 1, 2, ... He begins by putting cards 0 and 1 into a hat and drawing randomly to determine \( A_{1} \). The card assigned \( A_{1} \) does not get reused. Chebby then adds in card 2 and draws for \( A_{2} \), and continues in this manner to determine the random number. At each step, he only draws one card from two in the hat. Unfortunately, this method does not result in a uniform distribution. What is the expected value of Chebby's final number?

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