
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 $f(x) = a\ln x - x^2 + \frac{1}{2}a$ ($a \in \mathbb{R}$).
$(1)$ Discuss the monotonicity of the function $f(x)$;
$(2)$ If the function $f(x)$ always satisfies $f(x) \leqslant 0$ within its domain, find the range of the real number $a$.

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