
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 that the domain of the function $f(x)$ is $D$, if for all $x \in D$ there exists a unique $y \in D$ such that $\frac{f(x) + f(y)}{2} = C$ (where $C$ is a constant) holds true, then the function $f(x)$ is said to have an "average" of $C$ on $D$. Among the following four functions:
1. $f(x) = x^3 (x \in \mathbb{R})$;
2. $f(x) = (\frac{1}{2})^x (x \in \mathbb{R})$;
3. $f(x) = \ln{x} (x \in (0, +\infty))$;
4. $f(x) = 2\sin{x} (x \in \mathbb{R})$;
Find the function(s) whose "average" on their respective domains is $1$. (Fill in the serial number(s) of all functions that meet the condition.)

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