
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 $f(x)=\ln x-\frac{1}{2}ax^2$, $g(x)=e^x-bx$, where $a$, $b\in \mathbb{R}$. It is known that the tangent line of the curve $y=f(x)$ at the point $(1,f(1))$ is perpendicular to the line $x-y+1=0$. Find the values of $a$ and $b$.

$(Ⅰ)$ Find the value of $a$.

$(Ⅱ)$ Determine the monotonic intervals of $g(x)$.

$(Ⅲ)$ If $bf(x)+bx\leq xg(x)$ holds for all $x\in (0,+\infty)$, find the range of values for $b$.

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