
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 a sequence $\{a_n\}$, with the sum of its first $n$ terms denoted as $S_n$. If the slope of the tangent line at $x=a_n$ for the function $y=x^2-2x$ equals $S_n$, and a sequence $\{b_n\}$ satisfies that the point $(n,b_n)$ (where $n\in\mathbb{N}^*)$ lies on the line $y=x$.  
$(1)$ Find the general formulas for $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Calculate the sum of the first $n$ terms $T_n$ of the sequence $\{a_nb_n\}$.

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