
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 the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$. If $\frac{1}{2}≤\frac{{a}_{n+1}}{{a}_{n}}≤2(n∈{N}^{*})$, then the sequence $\{a_{n}\}$ is called a "compact sequence".<br/>$(1)$ If ${a}_{n}=\frac{{n}^{2}+2n}{{4}^{n}}$, determine whether $\{a_{n}\}$ is a "compact sequence", and explain the reason;<br/>$(2)$ If the sequence $\{a_{n}\}$ has a sum of the first $n$ terms as ${S}_{n}=\frac{1}{4}({n}^{2}+3n)$, determine whether $\{a_{n}\}$ is a "compact sequence", and explain the reason;<br/>$(3)$ Suppose the sequence $\{a_{n}\}$ is a geometric sequence with a common ratio of $q$. If both the sequences $\{a_{n}\}$ and $\{S_{n}\}$ are "compact sequences", find the range of values for $q$.

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