
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 plane rectangular coordinate system $xOy$, the equation of the line $l$ is $\sqrt{3}x-y+2\sqrt{3}=0$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $\rho =2\left( \sin \theta +\cos \theta \right)$.

1. Find the Cartesian equation of the curve $C$.
2. A perpendicular line is drawn from point $P(1,0)$ to line $l$, intersecting the curve $C$ at points $M$ and $N$. Find the value of $\frac{1}{|PM|}+\frac{1}{|PN|}$.

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