
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
It is known that the point $(1,2)$ lies on the graph of the function $f(x) = a^x$ (where $a>0$ and $a\neq 1$). The sum of the first $n$ terms of the sequence $\{a_n\}$ is given by $S_n = f(n) - 1$.
(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$;
(2) Let $b_n = \log_a (a_{n+1})$, find the sum of the first $n$ terms of the sequence $\{a_n b_n\}$, denoted as $T_n$.

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