Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, and $c$, we denote by $f(a, b, c)$ the number of ordered $6$-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions:
- (i) $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$;
- (ii) $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$;
- (iii) $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$.
Prove that $f(a,b,c)$ does not change when $a$, $b$, and $c$ are rearranged.
I tried drawing the venn diagram of the sets but I couldn't go anywhere from there. This is a problem from International Zhautykov Olympiad 2014. I hope someone could help me.