
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
Three persons \( A, B, C \) are playing the following game: A \( k \)-element subset of the set \(\{1, \ldots, 1986\}\) is randomly chosen, with an equal probability of each choice, where \( k \) is a fixed positive integer less than or equal to 1986. The winner is \( A, B \), or \( C \), respectively, if the sum of the chosen numbers leaves a remainder of 0, 1, or 2 when divided by 3. For what values of \( k \) is this game a fair one? (A game is fair if the three outcomes are equally probable.)

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