
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 grid square of size $n \times n$ where $n > 1$, we define a crossword as any non-empty set of its cells. A word is any horizontal or vertical strip (a grid rectangle with a width of one cell) consisting entirely of cells of the crossword and not contained in any larger strip of cells of the crossword (neither horizontal nor vertical). Let $x$ be the number of words in the crossword, and $y$ the minimum number of words needed to cover the crossword. Find the maximum ratio $\frac{x}{y}$ for a given $n$.

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