
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
The function $f(x)$ is defined as follows: for any real number $x$, if there exists an integer $m$ such that $|x-m| < \frac{1}{2}$, then $f(x)=m$. Given that the first term $a_{1}=1$ of the geometric sequence $\{a_{n}\}$, and $f(a_{2})+f(a_{3})=2$, determine the range of the common ratio $q$.

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