
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
Circle  $C$  with radius  $2$  has diameter  $\overline{AB}$ . Circle  $D$  is internally tangent to circle  $C$  at  $A$ . Circle  $E$  is internally tangent to circle  $C,$  externally tangent to circle  $D,$  and tangent to  $\overline{AB}$ . The radius of circle  $D$  is three times the radius of circle  $E$  and can be written in the form  $\sqrt{m} - n,$  where  $m$  and  $n$  are positive integers. Find  $m+n$ .

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