
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n = 2a_n - 2$, and the sequence $\{b_n\}$ is an arithmetic sequence with the first term $a_1$ and nonzero common difference, such that $b_1$, $b_3$, and $b_{11}$ form a geometric sequence.
(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;
(2) Let the sequence $\{c_n\}$ satisfy $c_n = \frac{1}{b_n b_{n+1}}$, and the sum of the first $n$ terms is $T_n$. If for any $n \in \mathbb{N}^+$ the inequality $T_n < t$ always holds, find the range of the real number $t$.

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