
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
Determine the locus of the midpoints of all chords of a given circle passing through a fixed point inside the circle, such that the distance from that point to the center of the circle is strictly greater than one half the radius and less than the radius.

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