
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 $C: \rho = 2\cos \theta$ is given. If we translate points on curve $C$ one unit to the left and then stretch the x-coordinate by a factor of 2 while keeping the y-coordinate unchanged, we obtain curve $C_1$. It is also known that the line $l: \begin{cases} y = \sqrt{3} + t\sin \frac{\pi}{3} \\ x = t\cos \frac{\pi}{3} \end{cases} (t$ is a parameter) intersects curve $C_1$ at points $A$ and $B$. Given point $P(0, \sqrt{3})$, find the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$.

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