
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

Let \( \mathbb{N} \) be the set of all positive integers. For a subset \( S \) of \( \mathbb{N} \) and \( n \in \mathbb{N} \), define
$$
S \oplus \{n\} = \{ s + n \mid s \in S \}.
$$

Additionally, define the subset \( S_k \) as follows:
$$
S_1 = \{1\}, \quad S_k = \left\{ S_{k-1} \oplus \{k\} \right\} \cup \{2k-1\}, \quad k = 2, 3, 4, \ldots 
$$

(1) Find \( \mathbb{N} - \bigcup_{k=1}^{\infty} S_k \).

(2) Find all \( k \in \mathbb{N} \) such that \( 1994 \in S_k \).

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