
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

A rigid square plate \(ABCD\) and a sphere \(G\) touching this plate at point \(E\), which is the intersection of the plate's diagonals, form a rigid system together. The system is supported on the horizontal plane \(S\) by the edge \(AB\) of the square plate and an appropriate point on the sphere.

The system is rotated about the vertex \(B\) in such a way that \(G\) continuously touches \(S\) until the edge \(BC\) reaches the plane \(S\). From this position, it is then similarly rotated around vertex \(C\) until edge \(CD\) reaches the plane \(S\), and so forth.

What trace does this system leave on the plane \(S\)? Given numerical data: the side length of the square is 4 units, and the radius of the sphere is 1 unit.

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