
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 functions $f(x) = x + \frac{a^2}{x}$ and $g(x) = -x - \ln(-x)$ where $a \neq 0$,
1. If $x = 1$ is an extremum of $f(x)$, find the value of the real number $a$ and the intervals of monotonicity for $g(x)$.
2. If for any $x_1 \in [1, 2]$, there exists $x_2 \in [-3, -2]$ such that $f(x_1) \geq g(x_2)$ always holds, and $-2 < a < 0$, determine the range of the real number $a$.

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