
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 of positive terms $\{a_n\}$ that satisfies $a_1=1$ and $a_n^2+a_n=a_{n+1}^2-a_{n+1}$, the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $S_n$, is given by $S_n=n^2+a_n$.

1. Find the general term formulas for the sequences $\{a_n\}$ and $\{b_n\}$.
2. Find the sum of the first $n$ terms, denoted by $T_n$, for the sequence $\{\frac{1}{a_{n+1}b_n}\}$.

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