
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 $f(x) = x^2 - \frac{a}{2}\ln x$ has a tangent line with a slope of $0$ at the point $\left( \frac{1}{2}, f\left( \frac{1}{2} \right) \right)$.

(Ⅰ) Discuss the monotonicity of $f(x)$;

(Ⅱ) If $g(x) = f(x) + \frac{1}{2}mx$ has no zeros on the interval $(1, +\infty)$, find the range of the real number $m$.

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