
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 $\{a\_n\}$ where $S\_n$ represents the sum of its first $n$ terms. If $\frac{1}{2} \leq \frac{a_{n+1}}{a\_n} \leq 2$ for all $n \in \mathbb{N}^*$, then $\{a\_n\}$ is called a "tight sequence".

(1) If $\{a\_n\}$ is a "tight sequence" with $a\_1=1, a\_2=\frac{3}{2}, a\_3=x, a\_4=4$, find the range of values for $x$.

(2) If $\{a\_n\}$ is an arithmetic sequence with first term $a\_1$ and common difference $d$ where $0 < d \leq a\_1$, determine whether $\{a\_n\}$ is a "tight sequence".

(3) Given that $\{a\_n\}$ is a geometric sequence with common ratio $q$. If both $\{a\_n\}$ and $\{S\_n\}$ are "tight sequences", find the range of values for $q$.

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