
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 \( S = \{1, 2, \cdots, 2006\} \) is divided into two disjoint subsets \( A \) and \( B \) such that:
1. \( B \subseteq A \);
2. If \( a \in A \), \( b \in B \), and \( a + b \in S \), then \( a + b \in B \);
3. If \( a \in A \), \( b \in B \), and \( ab \in S \), then \( ab \in A \).

Find the number of elements in subset \( A \).

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