
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 $f(x)=\ln x-ax^2$ at $x=1$, the tangent line is perpendicular to the line $x-y+1=0$.

(1) Find the monotonically increasing interval of the function $y=f(x)+xf'(x)$ (where $f'(x)$ is the derivative of $f(x)$).

(2) Let function $g(x)=f(x)+\frac{3}{2}x^2-(b+1)x$, and let $(x\_1, x\_2) (x\_1 < x\_2)$ be the two extreme points of the function $g(x)$. If $b\geqslant \frac{e^{2}+1}{e}-1.$ and $g(x\_1)-g(x\_2)\geqslant k$ always holds true, find the range of real numbers $k$.

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