
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 Cartesian coordinate system, the parametric equations of the line $C_{1}$ are $\left\{\begin{array}{l}x=1+t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ (where $t$ is the parameter). Using the origin $O$ as the pole and the positive x-axis as the polar axis, the polar equation of the curve $C_{2}$ is ${\rho}^{2}=\frac{4}{3-\cos2\theta}$.  
$(1)$ Find the Cartesian equation of the curve $C_{2}$.  
$(2)$ If the line $C_{1}$ intersects the curve $C_{2}$ at points $A$ and $B$, and $P(1,0)$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

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