HMMT 2011 U23

Let $S$ be the set of points $(x, y, z)$ in $\mathbb{R}^{3}$ such that $x, y$, and $z$ are positive integers less than or equal to 100. Let $f$ be a bijective map between $S$ and the $\{1,2, \ldots, 1000000\}$ that satisfies the following property: if $x_{1} \leq x_{2}, y_{1} \leq y_{2}$, and $z_{1} \leq z_{2}$, then $f\left(x_{1}, y_{1}, z_{1}\right) \leq f\left(x_{2}, y_{2}, z_{2}\right)$. Define

$$\begin{aligned} A_{i} & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(i, j, k), \\ B_{i} & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(j, i, k), \\ \text { and } C_{i} & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(j, k, i) \end{aligned}$$

Determine the minimum value of $A_{i+1}-A_{i}+B_{j+1}-B_{j}+C_{k+1}-C_{k}$.