
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
The circus arena is illuminated by \( n \) different spotlights. Each spotlight illuminates a certain convex shape. It is known that if any one spotlight is turned off, the arena will still be fully illuminated, but if any two spotlights are turned off, the arena will not be fully illuminated. For which values of \( n \) is this possible?

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