
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 $\overrightarrow{n}=(2\cos x,\sqrt{3}\sin x)$ and $\overrightarrow{m}=(\cos x,2\cos x)$, let $f(x)=\overrightarrow{n}\cdot \overrightarrow{m}+a$.

(I) If $x\in[0,\dfrac{\pi }{2}]$ and $a=1$, find the maximum and minimum values of $f(x)$, as well as the values of $x$ that yield these extremes.

(II) If $x\in[0,\pi]$ and $a=-1$, find the range of $b$ such that the equation $f(x)=b$ has two distinct real roots $x_{1}$, $x_{2}$, and find the value of $x_{1}+x_{2}$.

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