
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 the function $f(x) = \frac{7x+5}{x+1}$, the sequence $\{a_n\}$ satisfies $2a_{n+1} - 2a_n + a_{n+1}a_n = 0$ with $a_n \neq 0$. For the sequence $\{b_n\}$, $b_1 = f(0)$ and $b_n = f(a_n - 1)$.  
(1) Find the general term formula for the sequence $\{a_n\}$.  
(2) Find the sum of the first $n$ terms, $S_n$, of the sequence $\{a_na_{n+1}\}$.  
(3) Find the sum of the first $n$ terms, $T_n$, of the sequence $\{|b_n|\}$.

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