
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((5,1.5),1.5),black+linewidth(.75)); draw((0,0)--(10,0)--(10,8)--cycle,black+linewidth(.75)); MP("A",(0,0),SW);MP("B",(10,0),SE);MP("C",(10,8),NE);MP("P",(5,1.5),NW); MP("10",(5,0),S);MP("8",(10,4),E);MP("12.5",(5,4.5),NW); MP("->",(6.5,1.5),E); dot((5,1.5)); [/asy]
The sides of $\triangle ABC$ have lengths $8, 10,$ and $12.5$. A circle with center $P$ and radius $1.5$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. As $P$ completes one trip and returns to its original position, how far has $P$ traveled?

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