
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 natural number. At first the cells of a table  $2n$  x  $2n$  are colored in white. Two players  $A$  and  $B$  play the following game. First is  $A$  who has to color  $m$  arbitrary cells in red and after that  $B$  chooses  $n$  rows and  $n$  columns and color their cells in black. Player  $A$  wins, if there is at least one red cell on the board. Find the least value of  $m$  for which  $A$  wins no matter how  $B$  plays.

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