
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
Vincent has a fair die with sides labeled  $1$  to  $6$ . He first rolls the die and records it on a piece of paper. Then, every second thereafter, he re-rolls the die. If Vincent rolls a different value than his previous roll, he records the value and continues rolling. If Vincent rolls the same value, he stops, does \emph{not} record his final roll, and computes the average of his previously recorded rolls. Given that Vincent first rolled a  $1$ , let  $E$  be the expected value of his result. There exist rational numbers  $r,s,t > 0$  such that  $E = r-s\ln t$   and  $t$  is not a perfect power. If  $r+s+t = \frac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed  by Sean Li*

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