
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
Emma's calculator has ten buttons: one for each digit  $1, 2, \ldots, 9$ , and one marked ``clear''. When Emma presses one of the buttons marked with a digit, that digit is appended to the right of the display. When she presses the ``clear'' button, the display is completely erased. If Emma starts with an empty display and presses five (not necessarily distinct) buttons at random, where all ten buttons have equal probability of being chosen, the expected value of the number produced is  $\frac{m}{n}$ , for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ . (Take an empty display to represent the number 0.)

*Proposed by Michael Tang*

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