
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
Circles  $\omega$  and  $\gamma$  are drawn such that  $\omega$  is internally tangent to  $\gamma$ , the distance between their centers are  $5$ , and the area inside of  $\gamma$  but outside of  $\omega$  is  $100\pi$ . What is the sum of the radii of the circles?

[asy]
size(3cm);
real lw=0.4, dr=0.3;
real r1=14, r2=9;
pair A=(0,0), B=(r1-r2,0);
draw(A--B,dashed);
draw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));
filldraw(circle(A,dr)); filldraw(circle(B,dr));
label(" $5$ ",(A+B)/2,dir(-90));
label(" $\gamma$ ",A+r1*dir(135),dir(135)); label(" $\omega$ ",B+r2*dir(135),dir(135));
[/asy]

*2020 CCA Math Bonanza Individual Round #2*

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