How many prime divisors does the number $1\cdot 2003 + 2\cdot 2002 + 3\cdot 2001 + \cdots + 2001 \cdot 3 + 2002 \cdot 2 + 2003 \cdot 1$ have? Fill in the blank with the number of prime divisors.

Let's think step by step and output the final answer within \boxed{}.
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