
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
Let $Q$ be an interior point of circle $L$ other than its center, where the distance from $Q$ to the center of $L$ is one third the radius of $L$. Consider all the chords of $L$ that pass through $Q$. What is the shape and nature of the locus of midpoints of these chords?
A) a complete circle with radius $\frac{r}{3}$
B) a complete circle with radius $\frac{r}{6}$
C) a circle missing a point with radius $\frac{r}{6}$
D) a semicircle with radius $\frac{r}{3}$
E) none of these

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