
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 $\{a_n\}$, the sum of its first $n$ terms $S_n$ 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 $\{a_n\}$.
2. Let $b_n=a_n^2+S_na_n$. If the sequence $\{b_n\}$ is a geometric sequence, find the value of $t$.
3. Under the condition of (2), let $c_n=4a_n+1$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. If the inequality $\frac {12k}{4+n-T_n}\geqslant 2n-7$ holds for any $n\in\mathbb{N}^*$, find the range of the real number $k$.

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