
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 $f(x)=a\ln x-bx^{2}$, where $a,b\in R$, and the curve $y=f(x)$ is tangent to the line $y=-\frac{{1}}{{2}}$ at $x=1$.

(1) Find the values of $a$ and $b$;

(2) Find the maximum value of $f(x)$ on $\left[ \frac{1}{e},e\right] $.

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