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

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