
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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 \(\{1, 2, \ldots, 1000000\}\) that satisfies the following property: if \( x_{1} \leq x_{2} \), \( y_{1} \leq y_{2} \), and \( z_{1} \leq z_{2} \), then \( f(x_{1}, y_{1}, z_{1}) \leq f(x_{2}, y_{2}, z_{2}) \). Define
\[
\begin{aligned}
A_{i} & = \sum_{j=1}^{100} \sum_{k=1}^{100} f(i, j, k), \\
B_{i} & = \sum_{j=1}^{100} \sum_{k=1}^{100} f(j, i, k), \\
C_{i} & = \sum_{j=1}^{100} \sum_{k=1}^{100} f(j, k, i).
\end{aligned}
\]

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

Present the answer in LaTex format: \boxed{Your answer}
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