
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
Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of  $k$  colors, in such a way that for any  $k$  of the ten points, there are  $k$  segments each joining two of them and no two being painted with the same color. Determine all integers  $k$ ,  $1\leq k\leq 10$ , for which this is possible.

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