
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
You drop a 7 cm long piece of mechanical pencil lead on the ﬂoor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is  $\frac{m}n$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m + n$ .

*Proposed by Aaron Lin*

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