
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
Let the inverse function of $y=\ln x$ be $y=g(x)$, and let $f(x)=\frac{x^2}{e}\cdot g(x)-\frac{1}{3}x^3-x^2$ for $x \in \mathbb{R}$.
(Ⅰ) Determine the monotonic intervals of the function $y=f(x)$.
(Ⅱ) Find the minimum value of $y=f(x)$ on the interval $[-1, 2\ln(3)]$.

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