<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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  $100m+n$ . Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
902
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  $100m+n$ .
test