
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  $ABCD$  be a rectangle with:  $AB=a$ ,  $BC=b$ . Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides  $AB$  and  $AD$ ,the other is tangent to the sides  $CB$  and  $CD$ .

1. Find the distance between the centers of the circles(using  $a$  and  $b$ ).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

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