
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 we wish to pick a random integer between  $1$  and  $N$  inclusive by flipping a fair coin. One way we can do this is through generating a random binary decimal between  $0$  and  $1$ , then multiplying the result by  $N$  and taking the ceiling. However, this would take an infinite amount of time. We therefore stopping the flipping process after we have enough flips to determine the ceiling of the number. For instance, if  $N=3$ , we could conclude that the number is  $2$  after flipping  $.011_2$ , but  $.010_2$  is inconclusive.

Suppose  $N=2014$ . The expected number of flips for such a process is  $\frac{m}{n}$  where  $m$ ,  $n$  are relatively prime positive integers, find  $100m+n$ .

*Proposed by Lewis Chen*

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