
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 convex quadrilateral \(ABCD\) undergoes the following operation: one of its vertices is replaced by a point symmetric to this vertex with respect to the perpendicular bisector of the diagonal (not involving this vertex), marking the new point with the original letter. This operation is sequentially applied to the vertices \(A, B, C, D, A, B, \ldots\) - a total of \(n\) times. We will call a quadrilateral permissible if its sides are pairwise distinct and it remains convex after any number of operations. Is there:

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

b) a number \(n_{0}\) such that any permissible quadrilateral becomes identical to the original after \(n = n_{0}\) operations?

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