
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
Which of the following statements are correct?

① When ${f'}(x_{0})=0$ for the function $y=f\left(x\right)$ at $x=x_{0}$, it is not a necessary and sufficient condition for "$x=x_{0}$ is a critical point of the function $y=f\left(x\right)$";

② If the curve $y=f\left(x\right)$ has a tangent line at the point $(x_{0}, f(x_{0}))$, then ${f'}(x_{0})$ must exist;

③ For non-zero vectors $\overrightarrow{a},\overrightarrow{b}$, "$\overrightarrow{a}∥\overrightarrow{b}$" is not a necessary and sufficient condition for "$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow 0$";

④ "$a \gt 1$" is a sufficient but not necessary condition for "$\frac{1}{a}＜1$".

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