
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
Given a squared grid of size \( n \times n \), where \( n \geq 1 \). A crossword is defined as any non-empty set of its cells, and a word is any horizontal or vertical strip (a rectangular grid one cell wide) entirely consisting of cells of the crossword and not contained in any larger strip (either horizontal or vertical) of crossword cells. Let \( x \) be the number of words in the crossword, and \( y \) be the minimum number of words needed to cover the crossword. Find the maximum value of the ratio \( \frac{x}{y} \) for a given \( n \geq 2 \).

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