For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as $$F_t(t)=F_{t_i}(x)^n$$ and the CDF of $x=\min(x_1,\cdots,x_N)$ as $$F_x(x)=1-(1-F_{x_i}(x))^n$$
When we have $X=\max(\min(\cdots),\cdots,\min(\cdots))$, we may combine above results if every entry in $\min(\cdots)$ are independent to each other in each $\min$ term.
However, I have some common terms in $\min(\cdots)$ terms. For example: $$X=\max(\min(x_1,x_2,x_3),\min(x_1,x_4,x_5),\min(x_5,x_6,x_7),\min(x_3,x_6,x_8))$$ I understand that there are correlations, but I am not sure how I can apply, specially for the general case.
Can someone guide me to derive $F_X(x)$?