The sum \( \sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1} \) can be expressed as \( \frac{a}{b} \) for relatively prime positive integers \( a \) and \( b \). Find the remainder when \( a+b \) is divided by \( 1000 \).

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