
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 $f(x) = 2x\ln x - 2ax^{2}$, where $a \in \mathbb{R}$.  
$(1)$ When $a = \frac{1}{2}$, find the interval where $f(x)$ is monotonically decreasing.  
$(2)$ If $f(x) \leq \frac{{f'(x)}}{2} - \ln x - 1$ holds for all $x \in (1, +\infty)$, find the range of real numbers for $a$.

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