
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 that  $X_1, X_2, \ldots$  are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let  $S=\sum_{i=1}^kX_i/2^i,$   where  $k$  is the least positive integer such that  $X_k<X_{k+1},$  or  $k=\infty$  if there is no such integer. Find the expected value of  $S.$ 

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