
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
Consider a circle with a radius of 6 units and a center at the origin (0, 0) in a coordinate plane. A point \( P \) is set at coordinates (8, 0). Determine the locus of the midpoint of the segment \( PQ \) as \( Q \) moves along the perimeter of the circle, if a homothety with a scale factor of \( \frac{1}{3} \) is applied instead of \( \frac{1}{2} \).
A) A circle with center (4, 0) and radius 3 units
B) A circle with center \((\frac{16}{3}, 0)\) and radius 2 units
C) A circle with center \((\frac{16}{3}, 0)\) and radius 3 units
D) A circle with center (4, 0) and radius 2 units

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