
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
A and B are opposite corners of an \(n \times n\) board, divided into \(n^2\) squares by lines parallel to the sides. In each square, the diagonal parallel to \(AB\) is drawn, so that the board is divided into \(2n^2\) small triangles. The board has \((n + 1)^2\) nodes and a large number of line segments, each of length 1 or \(\sqrt{2}\). A piece moves from \(A\) to \(B\) along the line segments. It never moves along the same segment twice and its path includes exactly two sides of every small triangle on the board. For which \(n\) is this possible?

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