
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
Let \( n \) be a positive integer. Consider the following game: Initially, \( k \) stones are distributed among the \( n^2 \) squares of an \( n \times n \) chessboard. A move consists of choosing a square containing at least as many stones as the number of its adjacent squares (two squares are adjacent if they share a common edge) and moving one stone from this square to each of its adjacent squares.

Determine all positive integers \( k \) such that:

(a) There is an initial configuration with \( k \) stones such that no move is possible.

(b) There is an initial configuration with \( k \) stones such that an infinite sequence of moves is possible.

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