
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
Given a circle \( k \) with a radius of \( 1 \text{ cm} \) in the plane and a line \( e \) which is \( 2 \text{ cm} \) from the center of the circle. Let \( A \) be an arbitrary fixed point in the closed half-plane defined by \( e \) that does not contain the circle \( k \). Consider all parallelograms \( ABCD \) such that vertex \( B \) lies on the circle \( k \) and vertex \( D \) lies on the line \( e \). What is the geometric locus of vertex \( C \) for these parallelograms? (Also consider degenerate parallelograms where different points on the line are symmetric with respect to the midpoint of segments \( AC \) and \( BD \)).

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