
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 that the sequence $\{a_{n}\}$ is a geometric sequence with a common ratio $q \gt 1$, the sum of the first three terms is $13$, and $a_{1}$, $a_{2}+2$, $a_{3}$ are exactly the first, third, and fifth terms of an arithmetic sequence $\{b_{n}\}$.  
$(Ⅰ)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(Ⅱ)$ Let the general formula for sequence $\{c_{n}\}$ be $c_{n}=\left\{\begin{array}{l}{{a}_{n}, \text{ for odd } n,}\\{{b}_{n}, \text{ for even } n,}\end{array}\right.$, where $n\in \mathbb{N}^{*}$. Find the sum of the first $2n+1$ terms of sequence $\{c_{n}\}$ denoted as $S_{2n+1}$;  
$(Ⅲ)$ Find $\sum_{i=1}^{n}\frac{(2{b}_{i}-4){a}_{i+1}-1}{{b}_{{a}_{i+1}+1}•{b}_{{a}_{i+2}+1}}$ for $n\in \mathbb{N}^{*}$.

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