
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  $n\geq 2$  be an integer. Players  $A$  and  $B$  play a game using  $n\times 2021$  grid of square unit cells. Firstly,  $A$  paints each cell either black of white.  $B$  places a piece in one of the cells in the uppermost row, and designates one of the cells in the lowermost row as the *goal*. Then,  $A$  repeats the following operation  $n-1$  times:
・When the cell with the piece is painted white,  $A$  moves the piece to the cell one below.
・Otherwise,  $A$  moves the piece to the next cell on the left or right, and then to the cell one below.
Find the minimum possible value of  $n$  such that  $A$  can always move a piece to the *goal*, regardless of  $B$ 's choice.

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