
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 starts with an empty whiteboard. He then repeatedly chooses one of the digits  $1, 2, \ldots, 9$  (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of  $25$ . Let  $E$  be the expected number of digits that Bob writes. If  $E = \frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , find  $100m+n$ .

*Proposed by Amogh Gaitonde*

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