
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 $\{a_{n}\}$ that satisfies $a_{n+1}-a_{n}=1$, and the sum of its first $5$ terms is $15$; sequence $\{b_{n}\}$ is a geometric sequence with $b_{1}=2$, $4b_{2}$, $2b_{3}$, $b_{4}$ forming an arithmetic sequence.<br/>$(Ⅰ)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(Ⅱ)$ Calculate $\sum _{i=1}^{n}{a}_{i}{b}_{n+1-i},\sum _{i=1}^{2n-1}(-1)^{i-1}{a}_{i}^{2}(n\in{N}^{*})$.

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