
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 $f(x)=a\ln x - bx^2$, the equation of the tangent line at the point $P(2, f(2))$ on the graph is $y=-3x+2\ln 2 +2$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ If the equation $f(x)+m=0$ has two distinct real roots within the interval $\left[\frac{1}{e},e\right]$, find the range of $m$ (where $e$ is the base of the natural logarithm).

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