
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 Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are: $\left\{{\begin{array}{l}{x=-2+2\cos\alpha}\\{y=2\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter, and $\alpha \in \left[0,\pi \right]$). Point $P$ is any point on curve $C_{1}$. If point $P$ is rotated clockwise around the origin by $\frac{\pi}{2}$ to point $Q$, the locus of point $Q$ is curve $C_{2}$.  
$(1)$ Using the origin $O$ as the pole and the non-negative $x$-axis as the polar axis, establish the polar coordinate system and find the polar coordinate equation of curve $C_{2}$.  
$(2)$ Given point $F(0,-1)$, the line $\sqrt{3}x-y-1=0$ intersects curve $C_{2}$ at points $A$ and $B$. Find the value of $|FA|+|FB|$.

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