
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 $f(x) = x\ln x + \frac{1}{2}x^{2}$, $x_{0}$ is the extremum point of the function $f(x)$. Consider the following propositions: $① 0 < x_{0} < \frac{1}{e}$; $② x_{0} > \frac{1}{e}$; $③ f(x_{0}) + x_{0} < 0$; $④ f(x_{0}) + x_{0} > 0$; Among these, the correct propositions are ______________. (Fill in all the correct proposition numbers)

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