
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
Define a function $f(x)$ on $(-\infty,0)\cup(0,+\infty)$. If for any given geometric sequence $\{a_n\}$, $\{f(a_n)\}$ is still a geometric sequence, then $f(x)$ is called a "geometric sequence preserving function". The following functions are defined on $(-\infty,0)\cup(0,+\infty)$:

1. $f(x)=x^{2}$;
2. $f(x)=2^{x}$;
3. $f\left( x \right)=\sqrt{|x|}$;
4. $f(x)=\ln |x|$.

Among them, the "geometric sequence preserving functions" are $\boxed{\text{C: }①③}$.

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