
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
The diagram below shows a  $1\times2\times10$  duct with  $2\times2\times2$  cubes attached to each end. The resulting object is empty, but the entire surface is solid sheet metal. A spider walks along the inside of the duct between the two marked corners. There are positive integers  $m$  and  $n$  so that the shortest path the spider could take has length  $\sqrt{m}+\sqrt{n}$ . Find  $m + n$ .

[asy]
size(150);
defaultpen(linewidth(1));
draw(origin--(43,0)--(61,20)--(18,20)--cycle--(0,-43)--(43,-43)--(43,0)^^(43,-43)--(61,-23)--(61,20));
draw((43,-43)--(133,57)--(90,57)--extension((90,57),(0,-43),(61,20),(18,20)));
draw((0,-43)--(0,-65)--(43,-65)--(43,-43)^^(43,-65)--(133,35)--(133,57));
draw((133,35)--(133,5)--(119.5,-10)--(119.5,20)^^(119.5,-10)--extension((119.5,-10),(100,-10),(43,-65),(133,35)));
dot(origin^^(133,5));
[/asy]

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