
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 rectangular table of size \(14 \times n\) is given. A piece is initially placed in the top-left cell. Sasha and Olya take turns moving the piece one cell to the right or downward (Sasha moves first). A turn is considered a "pivot" if the direction of the move differs from the direction the opponent used in the previous turn. Sasha wins if the number of pivots is even by the time the piece reaches the bottom-right cell, otherwise, Olya wins. For which values of \(n\) can Olya guarantee a win regardless of Sasha's moves?

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