
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 $f(x)=(x+b)\cdot \ln x$, it is known that the tangent line to the curve $y=f(x)$ at the point $(1, f(x))$ is perpendicular to the line $x+2y=0$.

(Ⅰ) Find the value of $b$;

(Ⅱ) If the function $g(x)=e^{x}\cdot \left(\frac{f(x)}{x+1}-a\right)$ $(a\neq 0)$, 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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