
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 and B play a game on a $2^{100} \times 100$ grid. They take turns filling in symbols in the first row, with A going first. In each round, A chooses an empty cell in the first row and fills it with an "$\times$"; then, B chooses an empty cell in the first row and fills it with an "○". Once all the cells in the first row are filled, they start filling in the symbols in the second row, and so on, until all the cells in the grid are filled.

A's goal is to have as many distinct rows as possible in the grid, while B's goal is to have as few distinct rows as possible. If both use their optimal strategies, how many distinct rows will there be in the grid in the end?

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