
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 $f(x) = x^2 - ax$, $g(x) = \ln x$, and $h(x) = f(x) + g(x)$.

(I) If the monotonically decreasing interval of the function $y = h(x)$ is $(\frac{1}{2}, 1)$, find the value of the real number $a$;

(II) If $f(x) \geq g(x)$ holds true for any $x$ in the domain, find the range of values for the real number $a$;

(III) Assume that the function $y = h(x)$ has two extreme points $(x_1, x_2)$, and $x_1 \in (0, \frac{1}{2})$. If $h(x_1) - h(x_2) > m$ holds true, find the maximum value of the real number $m$.

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