
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 function $f(x)=(x+b)\ln x$, it is known that the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is perpendicular to the line $x+2y=0$.  
(I) Find the value of $b$.  
(II) If the function $g(x)=e^{x}\left( \frac {f(x)}{x+1}-a\right)(a\neq 0)$, and $g(x)$ is a monotonic function on the interval $(0,+\infty)$, find the range of the real number $a$.

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