
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 first $n$ terms sum, $S_n$, satisfies $S_n = t(S_n - a_n + 1)$, where $t$ is a constant and $t \neq 0, t \neq 1$.

(1) Find the general formula for the $n$-th term of the sequence.

(2) Let $b_n = a_n^2 + S_n \cdot a_n$. If the sequence $\{b_n\}$ is a geometric sequence, find the value of $t$.

(3) Under the conditions of part (2), let $c_n = 4a_n + 1$. If the sum of the first $n$ terms of the sequence $\{c_n\}$ is $T_n$, and the inequality $\frac{12k}{4 + n - T_n} \geq 2n - 7$ holds for all $n \in \mathbb{N}^*$, find the range of the real number $k$.

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