
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
Given a set  $S \subset N$  and a positive integer n, let  $S\oplus \{n\} = \{s+n / s \in  S\}$ . The sequence  $S_k$  of sets is defined inductively as follows:  $S_1 = {1}$ ,  $S_k=(S_{k-1} \oplus \{k\})  \cup \{2k-1\}$  for  $k = 2,3,4, ...$ (a) Determine  $N - \cup _{k=1}^{\infty}  S_k$ . 
(b) Find all  $n$  for which  $1994 \in S_n$ .

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