
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 an equilateral triangle with sidelength  $k$  cm. With lines parallel to it's sides, we split it into  $k^2$  small equilateral  triangles with sidelength  $1$  cm. This way, a triangular grid is created. In every small triangle of sidelength  $1$  cm,  we place exactly one integer from  $1$  to  $k^2$  (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths  $1$  cm. We shall name as *value* of the hexagon, the sum of the numbers that lie on the  $6$  small equilateral triangles that the hexagon consists of . Find (in terms of the integer  $k>4$ ) the maximum and the minimum value of the sum of the values of all hexagons .

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