
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 $f(x)=x\cos x+a$, where $a\in\mathbb{R}$.

(Ⅰ) Find the slope of the tangent line to the curve $y=f(x)$ at the point $x= \frac{\pi}{2}$.

(Ⅱ) Determine the number of roots of the equation $f'(x)=0$ ($f'(x)$ is the derivative of $f(x)$) in the interval $(0,1)$, and explain the reason.

(Ⅲ) If the function $F(x)=x\sin x+\cos x+ax$ has exactly one extreme point in the interval $(0,1)$, find the range of values for $a$.

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