
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 $y=\begin{cases} -x^{3}+x^{2},x<e\\a\ln x,x\geqslant e \end{cases}$, there exist two points $P$ and $Q$ on its graph such that $\triangle POQ$ is a right triangle with $O$ as its right vertex (where $O$ is the origin), and the midpoint of the hypotenuse is precisely on the $y$-axis. Determine the range of values for the real number $a$.

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