
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 \( A \) and \( B \) be two sets of \( N \) consecutive integers. If \( N = 2003 \), can we form \( N \) pairs \( (a, b) \) with \( a \in A \), \( b \in B \) such that the sums of the pairs are \( N \) consecutive integers? What about \( N = 2004 \)?

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