
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
Triangle  $ABC$  is equilateral with side length  $12$ . Point  $D$  is the midpoint of side  $\overline{BC}$ . Circles  $A$  and  $D$  intersect at the midpoints of side  $AB$  and  $AC$ . Point  $E$  lies on segment  $\overline{AD}$  and circle  $E$  is tangent to circles  $A$  and  $D$ . Compute the radius of circle  $E$ .

*2015 CCA Math Bonanza Individual Round #5*

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