
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 National Marriage Council wishes to invite  $n$  couples to form 17 discussion groups under the following conditions:

(1) All members of a group must be of the same sex; i.e. they are either all male or all female.

(2) The difference in the size of any two groups is 0 or 1.

(3) All groups have at least 1 member.

(4) Each person must belong to one and only one group.

Find all values of  $n$ ,  $n \leq 1996$ , for which this is possible.  Justify your answer.

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