
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
Let  $n$  be a positive integer. A square  $ABCD$  is partitioned into  $n^2$  unit squares. Each of them is divided into two triangles by the diagonal parallel to  $BD$ . Some of the vertices of the unit squares are colored red in such a way that each of these  $2n^2$  triangles contains at least one red vertex. Find the least number of red vertices.

*(4th Middle European Mathematical Olympiad, Team Competition, Problem 4)*

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