
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
Consider the function $f(x) = -x^2 + ax + 2(x^2 - x)\ln x$.
(Ⅰ) When $a=2$, determine the intervals on which $f(x)$ is monotonic.
(Ⅱ) Assume that for $x\in(0, +\infty)$, the inequality $f(x) + x^2 > 0$ holds true for all $x$. Find the minimum integer value of $a$.

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