A limit with respect to the WZ-pairs $\lim_{N \to \infty} \sum_{k=0}^{\infty} F(N+a,k+b)$

Question

Consider $$ F(n,k)=\frac{\left ( n+\frac12 \right )_k(3n)_k}{\left ( 2n+k \right ) (n+1)_k\left ( 3n+\frac12 \right )_k } \frac{4^n\,\Gamma(3n)\Gamma\left ( n+\frac12 \right )^4}{\Gamma\left ( n \right )^2\Gamma(2n+1)\Gamma\left ( 3n+\frac12 \right ) }, $$ where $(a)_k$ denotes the Pochhammer Symbols. One can easily check that its WZ-mate is $G(n,k)$: $$ G(n,k)=-\frac{k (8 k^3 n+2 k^3+64 k^2 n^2+32 k^2 n+3 k^2+168 k n^3+126 k n^2+25 k n+k+144 n^4+144 n^3+46 n^2+5 n)}{2 n^2 (k+2 n+1) (2 k+6 n+1) (2 k+6 n+3)}F(n,k) $$ such that $F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k).$ And they satisfty $$ \lim_{N \to \infty} \sum_{k=0}^{\infty} F(N+a,k+b)= \sum_{k\ge0}F(a,k+b)-\sum_{n\ge0}G(n+a,b) $$ for $a,b$ letting all the series convergent. But I am now have difficulty in calculating the limit. It should be simple, for which $\sum_{k\ge0}F(n,k) =\frac{\pi^{3/2}}{3}$ for arbitrary $n$. I am appreciated for your help.