
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
A rubber band is wrapped around two pipes as shown. One has radius  $3$  inches and the other has radius  $9$  inches. The length of the band can be expressed as  $a\pi+b\sqrt{c}$  where  $a$ ,  $b$ ,  $c$  are integers and  $c$  is square free. What is  $a+b+c$ ?

[asy]
size(4cm);
draw(circle((0,0),3));
draw(circle((12,0),9));
draw(3*dir(120)--(12,0)+9*dir(120));
draw(3*dir(240)--(12,0)+9*dir(240));
[/asy]

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