
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
Two concentric circles have radii  $1$  and  $4$ .  Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown.  The three lightly shaded circles are internally tangent to the circle with radius  $4$  while the three darkly shaded circles are externally tangent to the circle with radius  $1$ .  The radius of the six congruent circles can be written  $\textstyle\frac{k+\sqrt m}n$ , where  $k,m,$  and  $n$  are integers with  $k$  and  $n$  relatively prime.  Find  $k+m+n$ .

[asy]
size(150);
defaultpen(linewidth(0.8));
real r = (sqrt(133)-9)/2;
draw(circle(origin,1)^^circle(origin,4));
for(int i=0;i<=2;i=i+1)
{
filldraw(circle(dir(90 + i*120)*(4-r),r),gray);
}
for(int j=0;j<=2;j=j+1)
{
filldraw(circle(dir(30+j*120)*(1+r),r),darkgray);
}
[/asy]

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