Let \( K \) be the sum of the coefficients of the odd powers of \( x \) in the expansion of \( (1+x)^{99} \). Let \( a \) be the middle term in the expansion of \( \left( 2+\dfrac{1}{\sqrt{2}} \right)^{200} \). If \( \dfrac{{}^{200}C_{99}K}{a}=\dfrac{2^{\ell}m}{n} \), where \( m \) and \( n \) are odd numbers, then the ordered pair \( (\ell , n) \) is equal to:
