
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 a point \( H \) such that \( TH = TB_{1} \). Then,

\[ \angle THB_{1} = \angle TB_{1}H = \frac{\angle ATB_{1}}{2} = 60^{\circ} \]

Thus, \( HB_{1} = TB_{1} = B_{1}B \). Therefore,

\[ \angle BHB_{1} = \angle B_{1}BH = \frac{\angle TB_{1}H}{2} = 30^{\circ} \]

Therefore, \( \angle BHA = \angle BHB_{1} + \angle BHT = 90^{\circ} \). Since the perpendicular is shorter than the inclined line,

\[ AB > AH = AT + TH = AT + B_{1}T = AT + \frac{BT}{2} \]

Similarly, we establish the inequalities

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

Thus,

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

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