
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 \( BT \). Extend the segment \( AT \) beyond point \( T \) to a point \( H \) such that \( TH = TB_1 \). Then,

\[
\angle THB_1 = \angle TB_1H = \frac{\angle ATB_1}{2} = 60^{\circ}
\]

Therefore, \( HB_1 = TB_1 = B_1B \). Then,

\[
\angle BHB_1 = \angle B_1BH = \frac{\angle TB_1H}{2} = 30^{\circ}
\]

Thus, \( \angle BHA = \angle BHB_1 + \angle BHT = 90^{\circ} \). Since the perpendicular is shorter than the oblique line,

\[
\begin{aligned}
AB > AH & = AT + TH = \\
& = AT + B_1 T = AT + \frac{BT}{2}.
\end{aligned}
\]

Similarly, we establish the inequalities

\[
AC > AT + \frac{CT}{2} \quad \text{and} \quad BC > BT + \frac{CT}{2}
\]

\[
2 AB + 2 BC + 2 CA > 2\left(AT + \frac{BT}{2}\right) + 2\left(BT + \frac{CT}{2}\right) + 2\left(AT + \frac{CT}{2}\right) = 4 AT + 3 BT + 2 CT
\]

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