
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 circle centered at point  $A$  with radius  $19$  and the circle centered at point  $B$  with radius  $32$  are both internally tangent to a circle centered at point  $C$  with radius  $100$  such that point  $C$  lies on segment  $\overline{AB}$ . Point  $M$  is on the circle centered at  $A$  and point  $N$  is on the circle centered at  $B$  such that line  $MN$  is a common internal tangent of those two circles. Find the distance  $MN$ .
![Image](https://cdn.artofproblemsolving.com/attachments/3/d/1933ce259c229d49e21b9a2dcadddea2a6b404.png)

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