
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

On a $100 \times 100$ grid paper, the sheet is divided into $1 \times 2$ domino-shaped pieces. Two players take turns to carry out the following operation: A player connects any two squares that share a common edge and are not yet connected. If, after a player's move, the sheet forms a single connected piece (meaning the entire square can be lifted off the table by holding a single square), that player loses. Determine who is guaranteed to win with an optimal strategy: the starting player or their opponent?

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