
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 $\{a\_n\}$ where each term is positive, and the sum of the first $n$ terms is $S\_n$. The sequence satisfies the equation $a \_{n+1} ^{2} = 2S_{n} + n + 4$, and $a_{2} - 1$, $a_{3}$, $a_{7}$ are the first three terms of a geometric sequence $\{b\_n\}$.

1. Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$.
2. Let $c\_n = \frac{n}{b\_n} - \frac{1}{a\_n a\_{n+1}}$. Find the sum of the first $n$ terms, denoted as $T\_n$, of the sequence $\{c\_n\}$.

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