
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
The king promised a reward to anyone who could plant a beautiful fruit orchard in a rocky wasteland. Two brothers heard about this. The elder brother managed to dig 18 holes (see the left diagram). He could not dig more and broke all the shovels. The king became angry and imprisoned him. Then the younger brother, Ivan, proposed to place apple trees, pear trees, and plum trees at the vertices of equilateral triangles (see the right diagram), and to fill the remaining holes.

The king responded:
- "It will be nice if there are exactly three trees of each type growing at the vertices of equilateral triangles. But three types are too few. If you also plant apricots, I will release your brother. If you add a fifth type, cherries, I will pay you for the job. I also like almonds, but you won't be able to place six triangles here."
- "What if I can?"
- "Then ask for whatever you want!"

Ivan wondered if he could also get half the kingdom. Think about it: place as many types of trees as possible at the vertices of equilateral triangles. (Equality of triangles means that all sides and angles are equal, i.e., there is an exact match when superimposed; triangles can be rotated and flipped. Only one tree can grow in each hole.)

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