
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 that the point (1, 2) lies on the graph of the function $f(x) = a^x$ ($a>0$, $a \neq 1$), and the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = f(n) - 1$.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) If $b_n = \log_a a_{n+1}$, find the sum of the first $n$ terms, $T_n$, of the sequence $\{a_n \cdot b_n\}$.

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