
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  $k$  be a positive integer. Arnaldo and Bernaldo play a game in a table  $2020\times 2020$ , initially all the cells are empty. In each round a player chooses a empty cell and put one red token or one blue token, Arnaldo wins if in some moment, there are  $k$  consecutive cells in the same row or column with tokens of same color, if all the cells have a token and there aren't  $k$  consecutive cells(row or column) with same color, then Bernaldo wins. If the players play alternately and Arnaldo goes first, determine for which values of  $k$ , Arnaldo has the winning strategy.

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