
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 $f(x) = a\ln{x} + \frac{1}{x}$, where $a\in\mathbb{R}$.
1. If $f(x)$ has extremum values, find the range of $a$.
2. If $f(x)$ has a tangent line that passes through the origin, determine the range of $a$ and the number of such tangent lines, and provide the reasoning.

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