
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
Find all natural numbers \( k \) such that the set
\[ S=\{1994,1997,2000, \cdots, 1994+3k\} \]
can be partitioned into two disjoint sets \( A \) and \( B \) where the sum of all numbers in \( A \) is 9 times the sum of all numbers in \( B \).

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