
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
($A$) In the rectangular coordinate system $xOy$, the parametric equations of circle $C$ are given by $\begin{cases} & x=2\cos \varphi \ & y=2+2\sin \varphi \end{cases}(\varphi \text{ is the parameter})$, and the equation of line $l$ is $\sqrt{3}x+y-8=0$. Establish a polar coordinate system with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis.

(I) Find the polar equations of circle $C$ and line $l$;

(II) The ray $OT: \theta = \frac{\pi}{6}(\rho > 0)$ intersects circle $C$ at points $O$ and $M$, and intersects line $l$ at point $N$. Find the length of line segment $MN$.

($B$) Given the function $f(x)=|x-1|+|x-a|$.

(I) If $a=-2$, solve the inequality $f(x)\geqslant 5$;

(II) If $f(x)\geqslant 4$ always holds for all $x\in R$, find the range of values for the real number $a$.

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