
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
[asy] draw(circle((4,1),1),black+linewidth(.75)); draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); MP("A",(0,0),SW);MP("B",(8,0),SE);MP("C",(8,6),NE);MP("P",(4,1),NW); MP("10",(4,0),S);MP("8",(8,3),E);MP("12",(4,3),NW); MP("->",(5,1),E); dot((4,1)); [/asy]
The sides of \(\triangle ABC\) have lengths 10, 8, and 12. A circle with center \( P \) and radius 1 rolls around the inside of \(\triangle ABC\), always remaining tangent to at least one side of the triangle. When \( P \) first returns to its original position, through what distance has \( P \) traveled?

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