
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
A $100 \times 100$ checkered square is cut into dominoes. Two players play a game. On each turn, a player glues together two adjacent cells along a cut line. A player loses if, after their move, the shape becomes connected, meaning that the entire square can be lifted from the table by holding it with one cell. Who will win with perfect play – the starting player or their opponent?

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