If the sum of the first \( 20 \) terms of the series \(\displaystyle \frac{4 \cdot 1}{4 + 3 \cdot 1^{2} + 1^{4}} + \frac{4 \cdot 2}{4 + 3 \cdot 2^{2} + 2^{4}} + \frac{4 \cdot 3}{4 + 3 \cdot 3^{2} + 3^{4}} + \frac{4 \cdot 4}{4 + 3 \cdot 4^{2} + 4^{4}} + \ldots = \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to:
