
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  $\{1, . . . , 1986\}$  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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