
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  $\omega$  with radius  $1$  is given. A collection  $T$  of triangles is called *good*, if the following conditions hold:
[list=1]
[*] each triangle from  $T$  is inscribed in  $\omega$ ;
[*] no two triangles from  $T$  have a common interior point.
[/list]
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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