
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 $\{a_n\}$ be a sequence whose sum of the first $n$ terms is $S_n$. If the sequence satisfies $\frac {1}{2}\leq \frac {a_{n+1}}{a_{n}}\leq2$ for $n\in \mathbb{N^*}$, then $\{a_n\}$ 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 $x$.
2. If $\{a_n\}$ is an arithmetic sequence with the first term $a_1$ and common difference $d$, and $0<d\leq a_1$, determine whether $\{a_n\}$ is a "tight sequence".
3. Let $\{a_n\}$ be a geometric sequence with a common ratio $q$. If both $\{a_n\}$ and $\{S_n\}$ are "tight sequences," find the range of $q$.

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