
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 positive integer  $k$ , a pigeon and a seagull play a game on an  $n\times n$  board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose  $m$  grids and lay an egg in each grid he chooses. The seagull will choose a  $k\times k$  grids and eat all the eggs inside them. If at any point every grid in the  $n\times n $  board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer  $n\geq k$ , find all  $m$  such that the pigeon wins.

*Proposed by amano_hina*

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