
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  $m, n \ge 2$  be integers. Carl is given  $n$  marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points  $A$  and  $B$ , marks the  $m-1$  points that divide segment  $\overline{AB}$  into  $m$  congruent parts (but does not draw the segment).

For which pairs  $(m,n)$  can Carl necessarily accomplish his task, regardless of which  $n$  points he is given?

*Here, the *centroid* of  $n$  points with coordinates  $(x_1, y_1), \dots, (x_n, y_n)$  is the point with coordinates  $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$ .

*Proposed by Holden Mui and Carl Schildkraut*

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