
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
Bob is making partitions of  $10$ , but he hates even numbers, so he splits  $10$  up in a special way. He starts with  $10$ , and at each step he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example,  $6$  could be replaced with  $1+5$ ,  $2+4$ , or  $3+3$  all with equal probability. He terminates this process when all the numbers in his list are odd. The expected number of integers in his list at the end can be expressed in the form  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Compute  $100m+n$ .

*Proposed by Michael Ren*

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