
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
Let \( A_{1}, \ldots, A_{2022} \) be the vertices of a regular 2022-gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.) In each round, Bob chooses a point in the plane (not necessarily among \( A_{1}, \ldots, A_{2022} \)) and Alice responds truthfully with the color of that point. What is the smallest number \( Q \) for which Bob has a strategy to always determine the colors of points \( A_{1}, \ldots, A_{2022} \) in \( Q \) rounds?

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