
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 $f(x)=ax^2+bx$ whose graph passes through the point $(-1,0)$, and the slope of the tangent line at $x=-1$ is $-1$. Let the sequence $\{a_n\}$ have a sum of the first $n$ terms $S_n=f(n)$ for $n\in N^*$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Find the sum of the first $n$ terms $T_n$ for the sequence $\left\{ \frac{1}{a_n \cdot a_{n+1}} \right\}$.

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