
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
In the space are given  $2006$  distinct points, such that no  $4$  of them are coplanar. One draws a segment between each pair of points.
A natural number  $m$  is called *good* if one can put on each of these segments a positive integer not larger than  $m$ , so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a *good* number  $m$ .

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