
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 the first arithmetic progression be $a, a+d, a+2d, \ldots$, and the second arithmetic progression be $b, b+d, b+2d, \ldots$, where $d$ is the common difference of each sequence. Find the number of distinct numbers in the set $S$, which is the union of the first $2004$ terms of each sequence.

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