
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)--(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",(4,1),NW); 
MP("10",(5,0),S);MP("8",(10,4),E);MP("12.5",(5,4),NW); 
MP("->",(5,1),E); 
dot((4,1)); 
[/asy]
In $\triangle ABC$, the sides have lengths $8, 10,$ and $12.5$. 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}
user