
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 \( S_n \) be the sum of the first \( n \) terms of the sequence \( \{a_n\} \), such that \( a_1 + a_2 = 4 \) and \( \frac{2 S_{n+1} + 1}{2 S_n + 1} = \frac{a_2}{a_1} = c \) \((c > 0, n \in \mathbf{N}^*)\).

(1) Find the general term of the sequence \( \{a_n\} \).

(2) Let \( b_n = a_n \log_3 a_n \). Find the sum of the first \( n \) terms \( T_n \) for the sequence \( \{b_n\} \).

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