
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
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity  $v_1$ . The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity  $v_2$ . The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time  $v_3$ . The value of  $\frac{v_1}{v_3}+\frac{v_3}{v_1}$  can be expressed as a positive number  $\frac{m\sqrt{r}}{n}$ , where  $m$  and  $n$  are relatively prime, and  $r$  is not divisible by the square of any prime. Find  $m+n+r$ . If the number is rational, let  $r=1$ . 

*(Ahaan Rungta, 2 points)*

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