
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 $f(x) = ax + \frac{a-1}{x}$ ($a \in \mathbb{R}$), and $g(x) = \ln x$.
1. When $a=2$, find the minimum value of the function $h(x) = f(x) - g(x)$;
2. When $a > 0$, for any $x \geq 1$, the inequality $f(x) - g(x) \geq 1$ always holds, find the range of the real number $a$.

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