
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 a sequence of positive terms $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, it is known that $a_1 = 1$ and $S_{n+1} + S_n = a_{n+1}^2$. Another sequence $\{b_n\}$ satisfies $b_n \cdot b_{n+1} = 3^{a_n}$, with $b_1 = 1$.

1. Find the general terms for the sequences $\{a_n\}$ and $\{b_n\}$.
2. Denote $T_n = a_n b_2 + a_{n-1}b_4 + \cdots + a_1 b_{2n}$. Find $T_n$.

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