
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
The sides of a convex quadrilateral $ABCD$ are divided into $n \geq 2$ equal parts. Let the $k$-th division points from vertices $A$, $B$, $C$, and $D$ on sides $AB$, $BC$, $CD$, and $DA$ be denoted as $A_{k}$, $B_{k}$, $C_{k}$, $D_{k}$, respectively. For which pairs $(n, k)$ does the property hold that the quadrilateral $ABCD$ is a parallelogram if and only if the quadrilateral $A_{k}B_{k}C_{k}D_{k}$ is also a parallelogram?

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