
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 $xOy$, with the coordinate origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C_{1}$ is $\sqrt{2}\rho \cos (\theta -\frac{π}{4})+1=0$, and the parameter equation of curve $C_{2}$ is $\left\{\begin{array}{l}{x=2\cos\alpha}\\{y=\sqrt{3}\sin\alpha}\end{array}\right.$ ($\alpha$ is the parameter).  
$(1)$ Write down the Cartesian equation of curve $C_{1}$ and the general equation of curve $C_{2}$.  
$(2)$ Given the point $P(0,-1)$, curve $C_{1}$ intersects curve $C_{2}$ at points $A$ and $B$. Find the value of $\frac{1}{{|PA|}}+\frac{1}{{|PB|}}$.

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