
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 $f(x) = (x-1)^2$ and $g(x) = 4(x-1)$. For the sequence $\{a_n\}$, for any positive integer $n$, the equation $(a_{n+1} - a_n)g(a_n) + f(a_n) = 0$ holds, and $a_1 = 2$, when $n \geq 2$, $a_n \neq 1$; Let $b_n = a_n - 1$.

(Ⅰ) Find the general formula for the sequence $\{b_n\}$;

(Ⅱ) Let $S_n$ be the sum of the first $n$ terms of the sequence $\{nb_n\}$, $$T_{n}=S_{n}+ \frac {n\cdot 3^{n}}{4^{n-1}}+ \frac {3^{n}}{4^{n-2}}$$, find the value of $$\lim_{n \to \infty}T_{n}$$.

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