
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
Each cell of an  $100\times 100$  board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors?

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