
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 $2, 5, 8, \ldots$ and $7, 14, 21, \ldots$, and $10, 20, 30, \ldots$ be three arithmetic progressions. Find the number of distinct numbers in the union of the first $1500$ terms of each sequence.

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