
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
The sum of the first $n$ terms of the sequence $\{a_n\}$ is denoted by $S_n$, satisfying $a_1=1$ and $S_n - 2S_{n-1} = 1$ (for $n \in \mathbb{N}^*$ and $n \geq 2$). The sum of the first $n$ terms of the sequence $\{b_n\}$ is denoted by $T_n$, satisfying $b_1=1$ and $T_n = n^2b_n$ (for $n \in \mathbb{N}^*$).
(1) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$.
(2) If for $n \in \mathbb{N}^*$, it always holds that $S_n + 1 > \frac{\lambda}{b_n}$, find the range of the real number $\lambda$.

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