
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
For any interval  $\mathcal{A}$  in the real number line not containing zero, define its *reciprocal* to be the set of numbers of the form  $\frac 1x$  where  $x$  is an element in  $\mathcal{A}$ .  Compute the number of ordered pairs of positive integers  $(m,n)$  with  $m< n$  such that the length of the interval  $[m,n]$  is  $10^{10}$  times the length of its reciprocal.

*Proposed by David Altizio*

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