
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 $n$. A collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of \(\left(n^{2}-1\right) / 2\) 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 all possible values of $k(C)$ as a function of $n$.

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