
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
In the positive sequence $\{a_n\}$, $a_1 = 1$, and the point $A_n\left(\sqrt{a_n}, \sqrt{a_{n+1}}\right)$ lies on the hyperbola $y^2 - x^2 = 1$. For the sequence $\{b_n\}$, the point $(b_n, T_n)$ lies on the line $y=-\frac{1}{2}x + 1$, where $T_n$ is the sum of the first $n$ terms of the sequence $\{b_n\}$.  
(1) Find the general term formulas for the sequences $\{a_n\}$ and $\{b_n\}$, denoted as $a_n$ and $b_n$;  
(2) If $c_n = a_n \cdot b_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $S_n$.

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