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

(1) Find the general term formulas for sequences $\{a\_n\}$ and $\{b\_n\}$.
(2) If $c\_n=(-1)^{n}a\_n b\_n$, find the sum of the first $n$ terms, $T\_n$, for sequence $\{c\_n\}$.

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