
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 $f(x)=ax^2+b\ln x$ has an extreme value of $\frac{1}{2}$ at $x=1$:
$(1)$ Find the values of $a$ and $b$;
$(2)$ Determine the monotonicity of the function $y=f(x)$ and find the monotonic intervals.

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