
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
Two players, A and B, play a game on an $8 \times 8$ grid. Player A can color any two adjacent cells (sharing a common edge) black on their turn, while Player B can color any single cell white on their turn. Both players take turns, and initially, all cells are white. If Player B can, after their turn, ensure that:

1. Every $5 \times 5$ square on the grid has at least one white cell at one of its corners.
2. Every $5 \times 5$ square on the grid has at least two white cells at its corners.

Does Player B have a winning strategy?

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