
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
Given a finite set  $S \subset \mathbb{R}^3$ , define  $f(S)$  to be the mininum integer  $k$  such that there exist  $k$  planes that divide  $\mathbb{R}^3$  into a set of regions, where no region contains more than one point in  $S$ . Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Evaluate  $M(200) \cdot m(200)$ .

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