
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
Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, where $S_n=n^2+n$, and the general term of the sequence $\{b_n\}$ is given by $b_n=x^{n-1}$.

(1) Find the general term formula for the sequence $\{a_n\}$;

(2) Let $c_n=a_nb_n$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$.

   (i) Find $T_n$;

   (ii) If $x=2$, find the minimum value of the sequence $\left\{\dfrac{nT_{n+1}-2n}{T_{n+2}-2}\right\}$.

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