
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 $a\neq 0$, the function $f(x)= \begin{cases} -x^{3}+x^{2}, & x < e \\ a\ln x, & x\leqslant e \end{cases}$.
- $(1)$ Discuss the number of zeros of the function $f(x)$;
- $(2)$ If there exist two points $M$, $N$ on the graph of the function, such that $\triangle MON$ is a right-angled triangle with $O$ as the right-angle vertex (where $O$ is the origin), and the midpoint of the hypotenuse $MN$ is exactly on the $y$-axis, find the range of the real number $a$.

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