
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
From the sequence of positive integers \(1, 2, 3, 4, \cdots\), remove the multiples of 3 and 4, but keep all multiples of 5 (for example, \(15, 20, 60, 120\) should not be removed). After the removal, the remaining numbers form a new sequence in their original order: \(a_1 = 1, a_2 = 2, a_3 = 5, a_4 = 7, \cdots\). Find \(a_{2003}\).

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