
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
Given a circle with a circumference of $20$ units, find the number of possible values of the product $m \times n$ where $m$ and $n$ are the lengths of the two unequal arcs resulting from a secant drawn from a point outside the circle, given that the product of the lengths of two tangents drawn from the same point equals $m \times n$.

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