
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 set  $X$  has  $1983$  members. There exists a family of subsets  $\{S_1, S_2, \ldots , S_k \}$  such that:**(i)** the union of any three of these subsets is the entire set  $X$ , while**(ii)** the union of any two of them contains at most  $1979$  members. What is the largest possible value of  $k ?$ 

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