
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
On each side of a balance scale, $k$ weights, numbered from 1 to $k$, are placed, with the left side initially being heavier. It was found that if any two weights with the same number are swapped between the sides, the right side either becomes heavier or the scales balance. For which values of $k$ is this possible?

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