
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 grid square of size \( n \times n \) is given, where each square can be part of a strip. A strip is defined as any horizontal or vertical line segment (a grid rectangle of width one square), entirely made up of crossword cells and not contained within any larger strip (neither horizontal nor vertical). Let \( x \) be the number of strips in the crossword, and \( y \) be the minimum number of strips needed to cover the crossword. Find the maximum value of the ratio \( \frac{x}{y} \) for a given \( n \).

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