
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
Consider the function $f(x) = x^2 - 2ax + \ln x$ where $a \in \mathbb{R}$.
   (I) The tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ is perpendicular to the line $x - 2y + 1 = 0$. Find the value of $a$.
   (II) Discuss the monotonicity of the function $f(x)$.
   (III) The inequality $2x\ln x \geq -x^2 + ax - 3$ holds for all $x$ in the interval $(0, e]$. Determine the range of values for the real number $a$.

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