
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
Given a circle  $\omega$  with radius  $1$ . Let  $T$  be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set  $T$  is  $\omega$ ;
(b) The interior of any two triangles in the set  $T$  has no common point.
Find all positive real numbers  $t$ , for which for each positive integer  $n$  there is a good set of  $n$  triangles, where the perimeter of each triangle is greater than  $t$ .

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