
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
If a function $f(x)$ satisfies the condition that for any given geometric sequence ${a_n}$, ${f(a_n)}$ is still a geometric sequence, then $f(x)$ is called a "preserving geometric sequence function". Among the following functions:

1. $f(x) = 2x$
2. $f(x) = x + 1$
3. $f(x) = x^2$
4. $f(x) = 2^x$
5. $f(x) = \ln|x|$

Find the sequence numbers of all "preserving geometric sequence functions".

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