
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
Equilateral triangle  $ T$  is inscribed in circle  $ A$ , which has radius  $ 10$ . Circle  $ B$  with radius  $ 3$  is internally tangent to circle  $ A$  at one vertex of  $ T$ . Circles  $ C$  and  $ D$ , both with radius  $ 2$ , are internally tangent to circle  $ A$  at the other two vertices of  $ T$ . Circles  $ B$ ,  $ C$ , and  $ D$  are all externally tangent to circle  $ E$ , which has radius  $ \frac {m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m \plus{} n$ .
[asy]unitsize(2.2mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90);
pair Ep=(0,4-27/5);
pair[] dotted={A,B,C,D,Ep};

draw(Circle(A,10));
draw(Circle(B,3));
draw(Circle(C,2));
draw(Circle(D,2));
draw(Circle(Ep,27/5));

dot(dotted);
label(" $E$ ",Ep,E);
label(" $A$ ",A,W);
label(" $B$ ",B,W);
label(" $C$ ",C,W);
label(" $D$ ",D,E);[/asy]

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