
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 $f(x)=\ln x-ax^{2}$. The tangent line of the function at $x=1$ is perpendicular to the line $x-y+1=0$.
(I) Find the monotonically increasing interval of the function $y=f(x)+xf′(x)$ (where $f′(x)$ is the derivative of $f(x)$);
(II) Let $g(x)=f(x)+\frac{3}{2}x^{2}-(1+b)x$. Suppose $x_{1}$, $x_{2} (x_{1} < x_{2})$ are two extreme points of the function $g(x)$. If $b \geq \frac{e^{2}+1}{e}-1$, and $g(x_{1})-g(x_{2})\geq k$ always holds, find the maximum value of the real number $k$.

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