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 \(\left(2+\frac{1}{\sqrt{2}}\right)^{200}\). If \(\displaystyle \frac{\binom{200}{99} \, K}{a} = \frac{2^{\,\ell} \, m}{n}\), where \(m\) and \(n\) are odd numbers, then the ordered pair \((\ell,n)\) is: