
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
The ratio of the areas of two rectangles is $\frac{500}{125}$. After rationalizing the denominator, the ratio of one side length of the first rectangle to one side length of the second rectangle, given that both rectangles are similar and thus have proportional sides, can be expressed in simplified form $\frac{p\sqrt{q}}{r}$ where $p$, $q$, and $r$ are integers. What is the value of the sum $p+q+r$?

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