
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 $f\left(x\right)=\left\{\begin{array}{l}{1-{x}^{2}\ln x,x>0}\\{{e}^{-x-2},x\leq0}\end{array}\right.$.<br/>$(Ⅰ)$ When $x\in \left(0,+\infty \right)$, find the maximum value of $f\left(x\right)$;<br/>$(Ⅱ)$ If there exists $a\in \left[0,+\infty \right)$ such that the equation $f\left(x\right)+ax^{2}+bx=0$ has three distinct real roots, find the range of real number $b$.

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