
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 \( BT \). On the extension of segment \( AT \) beyond point \( T \), mark such a point \( H \) that \( TH = T B_1 \). Then
\[
\angle THB_1 = \angle TB_1H = \frac{\angle ATB_1}{2} = 60^\circ
\]
Therefore, \( HB_1 = TB_1 = B_1B \). Then
\[
\angle BHB_1 = \angle B_1B H = \frac{\angle TB_1H}{2} = 30^\circ
\]
Thus, \( \angle BHA = \angle BHB_1 + \angle BHT = 90^\circ \). Since the perpendicular is shorter than the inclined one,
\[
AB > AH = AT + TH = AT + B_1T = AT + \frac{BT}{2}
\]
Similarly, establish the inequalities
\[
AC > AT + \frac{CT}{2} \quad \text{and} \quad BC > BT + \frac{CT}{2}
\]
Thus,
\[
2AB + 2BC + 2CA > 2 \left( AT + \frac{BT}{2} \right) + 2 \left( BT + \frac{CT}{2} \right) + 2 \left( AT + \frac{CT}{2} \right) = 4AT + 3BT + 2CT
\]

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