
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:  
$①0 < x_{0} < \frac {1}{e}$；$②x_{0} > \frac {1}{e}$；$③f(x_{0})+x_{0} < 0$；$④f(x_{0})+x_{0} > 0$；  
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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