
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
With a convex quadrilateral \(ABCD\), the following operation is performed: one of the given vertices is changed to a point symmetric to this vertex with respect to the perpendicular bisector of the diagonal (not the end it belongs to), with the new point being denoted by the same letter as the original vertex. This operation is sequentially applied to the vertices \(A, B, C, D, A, B, \ldots\) - a total of \(n\) times. Let's call a quadrilateral permissible if its sides are pairwise distinct and it remains convex after any number of operations. Does there exist:

a) a permissible quadrilateral that becomes equal to the original after \(n<5\) operations;

b) a number \(n_0\) such that any permissible quadrilateral becomes equal to the original after \(n=n_0\) operations?

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