
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 $f(x) = e^x + a \ln x$ with a domain of $(0, +\infty)$, the following propositions are provided regarding the function $f(x)$:

$①$ When $a = 1$, $f(x)$ has a maximum value;

$②$ When $a = -1$, for any $x > 0$, $f(x) > 0$;

$③$ For any $a > 0$, $f(x)$ is an increasing function on $(0, +\infty)$;

$④$ For any $a < 0$, $f(x)$ definitely has a minimum value.

Write down the sequence number(s) of all the correct propositions.

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