
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

Let \( B_{1} \) be the midpoint of segment \( B T \). On the extension of segment \( A T \) beyond point \( T \), let us mark a point \( H \) such that \( T H = T B_{1} \). Then

$$
\angle T H B_{1} = \angle T B_{1} H = \frac{\angle A T B_{1}}{2} = 60^{\circ}
$$

Therefore, \( H B_{1} = T B_{1} = B_{1} B \). Then

$$
\angle B H B_{1} = \angle B_{1} B H = \frac{\angle T B_{1} H}{2} = 30^{\circ}
$$

Thus, \( \angle B H A = \angle B H B_{1} + \angle B H T = 90^{\circ} \). Since the perpendicular is shorter than the oblique,

$$
\begin{aligned}
A B > A H & = A T + T H = \\
& = A T + B_{1} T = A T + \frac{B T}{2}
\end{aligned}
$$

Similarly, establish the inequalities

$$
A C > A T + \frac{C T}{2} \quad \text{and} \quad B C > B T + \frac{C T}{2}
$$

Thus,

$$
\begin{aligned}
2 A B + 2 B C + 2 C A & > 2 \left(A T + \frac{B T}{2}\right) + 2 \left(B T + \frac{C T}{2}\right) + 2 \left(A T + \frac{C T}{2}\right) = \\
& = 4 A T + 3 B T + 2 C T
\end{aligned}
$$

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