
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 a geometric sequence with a common ratio greater than $0$, and the sum of its first $n$ terms is denoted as ${S}_{n}(n∈{N}^{*})$. Let $\{b_{n}\}$ be an arithmetic sequence. Given $a_{1}=1$, $a_{3}=a_{2}+2$, $a_{4}=b_{3}+b_{5}$, and $a_{5}=b_{4}+2b_{6}$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let the sum of the first $n$ terms of the sequence $\{S_{n}\}$ be denoted as ${T}_{n}(n∈{N}^{*})$.  
$(i)$ Find $T_{n}$.  
$(ii)$ Find $\sum_{k=1}^{n}{{(-1)}^{k}\frac{(3{b}_{k}+4)({T}_{k}+{b}_{k+2})}{(k+1)(k+2)}}$.

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