
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
In the polar coordinate system, the curve C is given by $\rho=2\cos\theta$. If points on curve C are shifted one unit to the left and then the x-coordinates are doubled while the y-coordinates remain unchanged, we obtain curve $C_1$. It is also known that 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 that line $l$ intersects curve $C_1$ at points A and B.  
(1) Find the rectangular coordinate equation of curve $C_1$ and identify what type of curve it is;  
(2) Let point P be $(0, \sqrt{3})$, calculate $\frac{1}{|PA|} + \frac{1}{|PB|}$.

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