
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 $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be two mutually tangent circles with a radius of 1. Let $T$ be their point of tangency, and $S$ be the point on $\mathcal{C}_{2}$ diametrically opposite to $T$. A point $A$ moves along the perimeter of $\mathcal{C}_{1}$ in a clockwise direction, starting from $T$. A point $B$ moves along the perimeter of $\mathcal{C}_{2}$ at the same speed as $A$, in a counterclockwise direction, starting from $S$. Describe the locus of the midpoints of the segments $[AB]$.

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