
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 circle \(\omega\) of radius 1 is given. A collection \(T\) of triangles is called good if the following conditions hold:
(i) each triangle from \(T\) is inscribed in \(\omega\);
(ii) no two triangles from \(T\) have a common interior point.

Determine all positive real numbers \(t\) such that, for each positive integer \(n\), there exists a good collection of \(n\) triangles, each of perimeter greater than \(t\).

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