
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
Alice and Bob have an $8 \times 8$ chessboard in front of them. Initially, all the squares are white. Each turn, Alice selects a white square and colors it black. Bob then chooses to color one of the neighboring squares (sharing an edge) black or does nothing. Alice can stop the game whenever she wants. Her goal is to maximize the number of black connected components, while Bob wants to minimize this number. If both players play optimally, how many connected components are there at the end of the game?

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