
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 the following conditions: $\left(1\right)y=f\left(x\right)$ intersects with $y=2x$; $\left(2\right)$ the derivative of the function $f\left(x\right)$ is ${f'}\left(x\right)$, and the values of ${f'}\left(x\right)$ are all within the interval $\left(0,2\right)$. We denote the set of functions that simultaneously satisfy the above two conditions in the domain as $U$. Which of the following statements are correct?  
① If $f\left(x\right)\in U$, then the equation $f\left(x\right)=2x$ has exactly one solution;  
② For the function $g\left(x\right)=2\ln x$, $\left(x \gt 1\right)$, $g\left(x\right)\in U$, but $g\left(x\right)+3\notin U$;  
③ Let $h\left(x\right)\in U$, for any $x_{1}$ and $x_{2}$ in the domain of $h\left(x\right)$ such that $|x_{1}-2023| \lt 1$ and $|x_{2}-2023| \lt 1$, then $|h(x_{1})-h(x_{2})| \lt 4$.

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