
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
Two points are located  $10$  units apart, and a circle is drawn with radius  $ r$  centered at one of the points. A tangent line to the circle is drawn from the other point. What value of  $ r$  maximizes the area of the triangle formed by the two points and the point of tangency?


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