
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
Given the sequence \( T_0 = 2 \), \( T_1 = 3 \), \( T_2 = 6 \), and for \( n \geq 3 \),
\[ T_n = (n+4) T_{n-1} - 4n T_{n-2} + (4n - 8) T_{n-3}, \]
the first \( n \) terms of this sequence are \( 2, 3, 6, 14, 40, 152, 784, 5186, 40576, \ldots \). Find the general term formula for \( T_n \) which is of the form \( T_n = A_n + B_n \), where \( \{A_n\} \) and \( \{B_n\} \) are well-known sequences, and provide a proof.

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