
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
For the function $f(x) = x\ln x$, we have the following conclusions:
1. The function is an even function;
2. If $f'(x_0) = 2$, then $x_0 = e$;
3. Its interval of monotonic increase is $[\frac{1}{e}, +\infty)$;
4. The range is $[\frac{1}{e}, +\infty)$;
5. The graph of the function and the line $y = -\frac{1}{e}$ have exactly one common point. (In this problem, $e$ represents the base of the natural logarithm.)
Fill in the correct conclusions' numbers on the line ______.

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