
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
On a blackboard lies  $50$  magnets in a line numbered from  $1$  to  $50$ , with different magnets containing different numbers.  David walks up to the blackboard and rearranges the magnets into some arbitrary order.  He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets.  If the expected number of times he writes the number  $1$  can be written in the form  $\tfrac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by David Altizio*

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