
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 game board composed of $n \times n$ square fields, where $n \geq 2$, fields that are directly horizontally or vertically adjacent to a field are called its neighbors. Initially, $k$ game pieces are distributed on the fields, where a field can contain multiple or no game pieces.

In each turn, one selects a field that has at least as many game pieces as neighbors and places one of these game pieces on each neighboring field. The game ends when there is no such field available for selection.

(a) Determine the smallest $k$ for which the game does not end for any possible initial arrangement and choice of moves.

(b) Determine the largest $k$ for which the game ends for any possible initial arrangement and choice of moves.

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