
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 $P$ be a point inside circle $K$ located at distance $d$ from the center $O$ of $K$, where $d$ is exactly one-third the radius $r$ of $K$. Consider all chords of $K$ that subtend an angle of $60^{\circ}$ at the center $O$ and pass through $P$. Determine the locus of the midpoints of these chords.

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