
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 $f(x) = -x^2 + a \ln x$ $(a \in \mathbb{R})$.
- (I) When $a = 2$, find the equation of the tangent line to the function $f(x)$ at the point $(1, f(1))$.
- (II) If $g(x) = f(x) - 2x + 2x^2$, discuss the monotonicity of the function $g(x)$.
- (III) If the function $g(x)$ in (II) has two extreme points $x_1, x_2$ $(x_1 < x_2)$, and the inequality $g(x_1) \geqslant mx_2$ always holds, find the range of the real number $m$.

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