
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 a function $f(x) = x^2 - a\ln x$, where $a$ is a real number.  
$(1)$ Discuss the monotonicity of the function $f(x)$.  
$(2)$ Let $g(x) = f(x) - x^2 - ax + 1 + (a + x)\ln x$. If $g(x) \geq 0$ always holds, find the range of real numbers for $a$.

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