
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
In 1859, the Chinese mathematician Li Shanlan of the Qing Dynasty translated the term "$function$" into "函数" (function) and gave the definition: "If a variable changes with another variable, then this is a function of that."

① If $f(-2) = f(2)$, then the function $f(x)$ is an even function.

② If the function $f(x)$ defined on $\mathbb{R}$ is monotonically increasing on the interval $(-\infty, 0]$ and monotonically increasing on the interval $[0, +\infty)$, then the function $f(x)$ is an increasing function on $\mathbb{R}$.

③ If the domain of the function $y = f(x)$ is $\left[a, b\right]$, where $a < c < b$, and if $f(x)$ is increasing on $\left[a, c\right)$ and decreasing on $\left[c, b\right]$, then $f(x)_{\max} = f(c)$.

④ For any $x_{1}, x_{2} \in (0, +\infty)$, the function $f(x) = \sqrt{x}$ satisfies $\frac{{f(x_{1}) + f(x_{2})}}{2} \leq f(\frac{{x_{1} + x_{2}}}{2})$.

Which of the following statements about the properties of functions are correct? (Please write down the serial numbers of all correct answers).

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