
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
There are $k$ weights numbered from 1 to $k$ on each of the two pans of a balance scale, with the left pan being heavier. If any two weights with the same number are swapped between the pans, the right pan always becomes heavier or the pans balance. Find all possible values of $k$.

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