
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 $f(x) = x^2 - px - q\ln x$ where $p, q \in \mathbb{R}$.  
$(1)$ If $p = a$, $q = a^2$, and the minimum value of $f(x)$ is $0$, find the value of the non-zero real number $a$.  
$(2)$ If $p = a^2$, $q = a$, and $f(x) \geqslant 0$ always holds, find the range of real numbers for $a$.

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