
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
Let $\{a_{n}\}$ be an arithmetic sequence with the sum of its first $n$ terms denoted as $S_{n}$, and let $\{b_{n}\}$ be a sequence of positive terms in geometric progression. Given that $a_{1}=b_{1}=2$, $S_{4}=a_{5}+b_{3}$, and $a_{3}+b_{2}=8$. Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$. If ____ in question (2), find the sum $T_{n}$ of the first $n$ terms of sequence $\{c_{n}\}$. Given the conditions $c_{n}=\frac{1}{{a_{n}{a_{n+1}}}}+{b_{n}}$, $c_{n}=a_{n}b_{n}$, and $c_{n}=\frac{{a_{n}+2}}{{a_{n}{a_{n+1}}{b_{n+1}}}$, choose any one of these conditions to complete question (2) and solve it.

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