
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 side lengths  $a,b,c$  of a triangle  $ABC$  are positive integers. Let:

\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]

for any positive integer  $n$ . If  $\frac{T_{2}}{2T_{1}}=2023$  and  $a>b>c$  , determine all possible perimeters of the triangle  $ABC$ .

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