
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
Choose one of the following two problems to answer. If more than one problem is done, only the first one will be graded.

(1) Elective 4-4: Coordinate Systems and Parametric Equations

In the polar coordinate system, the curve $C$ is given by the equation $p=2\cos\theta$. Translate the curve $C$ one unit to the left and then stretch the horizontal coordinate to twice its original length, resulting in the curve $C\_1$. The line $l$ is defined by the parametric equations $\begin{cases} x=t\cos\frac{\pi}{3} \ y=\sqrt{3}+t\sin\frac{\pi}{3} \end{cases}$ (where $t$ is a parameter), and it intersects the curve $C\_1$ at points $A$ and $B$.

(i) Find the rectangular equation of the curve $C\_1$ and identify the type of curve it represents.

(ii) Let point $P$ be $(0,\sqrt{3})$. Find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

(2) Elective 4-5: Inequalities

The function $f(x)=|x-a|$ is given, where $a\in\mathbb{R}$.

(i) When $a=1$, solve the inequality $f(x)<|2x-1|-1$.

(ii) When $x\in(-2,1)$, find the range of $a$ such that $|x-1|>|2x-a-1|-f(x)$.

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