
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
Let the function $f(x)=ax^2+bx+c\ln x$ where $a$, $b$, $c$ are real constants.

$(1)$ When $b=0,c=1$, discuss the monotonic intervals of $f(x)$;

$(2)$ The tangent line of the curve $y=f(x)$ (where $a > 0$) at the point $(1,f(1))$ is $y=3x-3$. If the function $f(x)$ has no extremum and $f'(x)$ has a zero point, find the values of $a$, $b$, $c$;

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