
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
Consider an \( n \)-by-\( n \) board of unit squares for some odd positive integer \( n \). We say that a collection \( C \) of identical dominoes is a maximal grid-aligned configuration on the board if \( C \) consists of \(\left(\frac{n^2 - 1}{2}\right)\) dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: \( C \) then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let \( k(C) \) be the number of distinct maximal grid-aligned configurations obtainable from \( C \) by repeatedly sliding dominoes. Find the maximum possible value of \( k(C) \) as a function of \( n \).

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