
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 rectangular sheet of paper, a circle is drawn, within which Misha mentally selects $n$ points, and Kolya tries to guess them. In one attempt, Kolya points to a spot on the sheet (inside or outside the circle), and Misha tells Kolya the distance from that spot to the nearest unguessed point. If the distance is zero, then the indicated point is considered guessed. Kolya can mark points on the sheet, measure distances, and perform constructions using a compass and a ruler. Can Kolya definitely guess all the chosen points in fewer than $(n+1)^{2}$ attempts?

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