
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
Let \(a \neq b\) be positive real numbers and \(m, n\) be positive integers. An \(m+n\)-gon \(P\) has the property that \(m\) sides have length \(a\) and \(n\) sides have length \(b\). Further suppose that \(P\) can be inscribed in a circle of radius \(a+b\). Compute the number of ordered pairs \((m, n)\), with \(m, n \leq 100\), for which such a polygon \(P\) exists for some distinct values of \(a\) and \(b\).

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