
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 $f(x) = x \ln x$, the following conclusions are drawn:

1. The function is an even function.
2. If $f'(x\_0) = 2$, then $x\_0 = e$.
3. Its monotonically increasing interval is $[\frac{1}{e}, +\infty)$.
4. Its range is $[\frac{1}{e}, +\infty)$.
5. The graph of the function intersects with the line $y = -\frac{1}{e}$ at exactly one point.

(In this question, $e$ denotes the base of the natural logarithm.)

Determine the correct conclusion(s) by filling in the blank with the corresponding number(s).

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