
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 cylinder radius  $12$  and a cylinder radius  $36$  are held tangent to each other with a tight band. The length of the band is  $m\sqrt{k}+n\pi$  where  $m$ ,  $k$ , and  $n$  are positive integers, and  $k$  is not divisible by the square of any prime. Find  $m + k + n$ .

[asy]
size(150);
real t=0.3;
void cyl(pair x, real r, real h)
{
pair xx=(x.x,t*x.y);
path 
B=ellipse(xx,r,t*r),
T=ellipse((x.x,t*x.y+h),r,t*r),
S=xx+(r,0)--xx+(r,h)--(xx+(-r,h))--xx-(r,0);
unfill(S--cycle); draw(S);
unfill(B); draw(B);
unfill(T); draw(T); 
}
real h=8, R=3,r=1.2;
pair X=(0,0), Y=(R+r)*dir(-50);
cyl(X,R,h);
draw(shift((0,5))*yscale(t)*arc(X,R,180,360));
cyl(Y,r,h);
void str (pair x, pair y, real R, real r, real h, real w)
{
real u=(angle(y-x)+asin((R-r)/(R+r)))*180/pi+270;
path P=yscale(t)*(arc(x,R,180,u)--arc(y,r,u,360));
path Q=shift((0,h))*P--shift((0,h+w))*reverse(P)--cycle;
fill(Q,grey);draw(Q);
}
str(X,Y,R,r,3.5,1.5);[/asy]

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