
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
Two circles with centers at $A$ and $B$ have $\overline{AB} = 50$. Points $P$ and $Q$ lie on both circles, and segment PQ is a common chord of length 40. The radius of circle $A$ is $\tfrac{2}{3}$ times the radius of circle $B$. Let $R$ be the midpoint of segment $\overline{PQ}$. Calculate $\overline{AR} + \overline{BR}$.

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