
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 function $f\left(x\right)=x+\sin x$, $\left(x\in R\right)$, the following four propositions are known:<br/>①For $\forall x \gt 0$, the inequality $f\left(x\right) \lt 2x$ always holds;<br/>②$x=\pi $ is a critical point of the function $f\left(x\right)$;<br/>③The area enclosed by the graph of the function $f\left(x\right)$, the x-axis, and $x=\frac{π}{2}$ is $\frac{{{π^2}}}{8}+1$;<br/>④$sin\frac{π}{7}+sin\frac{{6π}}{7}＜\frac{{2π}}{7}$.<br/>The correct propositions are______.

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