
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 snowman is built on a level plane by placing a ball radius  $6$  on top of a ball radius  $8$  on top of a ball radius  $10$  as shown. If the average height above the plane of a point in the snowman is  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers, find  $m + n$ .

[asy]
size(150);
draw(circle((0,0),24));
draw(ellipse((0,0),24,9));
draw(circle((0,-56),32));
draw(ellipse((0,-56),32,12));
draw(circle((0,-128),40));
draw(ellipse((0,-128),40,15));
[/asy]

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