
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 functions $f(x) = x + \frac{a^2}{x}$ and $g(x) = x + \ln(x)$, where $a > 0$.
(1) If $x = 1$ is an extremum point of the function $h(x) = f(x) + g(x)$, find the value of the real number $a$.
(2) If for any $x_1, x_2 \in (1, e)$ ($e$ is the base of the natural logarithm), the inequality $f(x_1) \geq g(x_2)$ holds, find the range of the real number $a$.

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