
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
A new operation is defined as: $\left|\begin{matrix}a & b \\ c & d\end{matrix}\right|=ad-bc$. If the sequence $\{a_n\}$ satisfies $a_1= \frac{2}{3}$ and $\left|\begin{matrix}a_{n+1} & n \\ a_{n} & n+1\end{matrix}\right|=0$.

① Find the general formula for the sequence $\{a_n\}$.

② If the sequence $\{b_n\}$ satisfies: $b_n=a_na_{n+1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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