
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
Points  $A$ ,  $B$ , and  $C$  lie in the plane such that  $AB=13$ ,  $BC=14$ , and  $CA=15$ .  A peculiar laser is fired from  $A$  perpendicular to  $\overline{BC}$ .  After bouncing off  $BC$ , it travels in a direction perpendicular to  $CA$ .  When it hits  $CA$ , it travels in a direction perpendicular to  $AB$ , and after hitting  $AB$  its new direction is perpendicular to  $BC$  again.  If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape  $T_\infty$ .  What is the ratio of the perimeter of  $T_\infty$  to the perimeter of  $\triangle ABC$ ?

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