
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
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, with points $P$ and $Q$ lying on all four circles. The radius of circle $A$ is $\frac{2}{3}$ the radius of circle $B$, and the radius of circle $C$ is $\frac{2}{3}$ the radius of circle $D$. Moreover, $AB = CD = 50$ and $PQ = 50$. Let $R$ be the midpoint of $\overline{PQ}$. Find $\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}$.
A) 50 B) 75 C) 100 D) 125 E) 150

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