
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 $f\left(x\right)=a\ln x-bx^{2}$, where $a$ and $b$ are real numbers. The function $f\left(x\right)$ is tangent to the line $y=-\frac{1}{2}$ at $x=1$.  
$(1)$ Find the values of the real numbers $a$ and $b$.  
$(2)$ Determine the monotonicity of the function $f\left(x\right)$ on the interval $[\frac{1}{e},e]$.

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