
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 $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. If $\frac{1}{2} \leq \frac{a_{n+1}}{a\_n} \leq 2$ for all $n \in \mathbb{N}^*$, the sequence is called a "tight sequence".
(1) If $a\_1 = 1$, $a\_2 = \frac{3}{2}$, $a\_3 = x$, and $a\_4 = 4$, find the range of possible 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) If $\{a\_n\}$ is a geometric sequence with common ratio $q$, and both $\{a\_n\}$ and $\{S\_n\}$ are "tight sequences", find the range of possible values for $q$.

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