
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
We are given a circle with center $O$ located at $(0,0)$ and radius 10 units. A point $P$ is placed at $(15, 0)$. There is a second smaller circle with a radius of 5 units also centered at $O$ which intersects the first circle at their southernmost and northernmost points. Determine the locus of the midpoint $M$ of the segment $PQ$ as point $Q$ varies over the larger circle.
A) Circle centered at $(7.5, 0)$ with radius 5 units
B) Circle centered at $(7.5, 0)$ with radius 10 units
C) Circle centered at $(15, 0)$ with radius 5 units
D) Circle centered at $(0, 0)$ with radius 7.5 units

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