
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 parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=t+\frac{1}{t}}\\{y=t-\frac{1}{t}}\end{array}}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $\rho =2\sin \theta$.  
$(1)$ Find the rectangular coordinate equations of curves $C_{1}$ and $C_{2}$.  
$(2)$ Let point $A$ be on curve $C_{1}$ and point $B$ be on curve $C_{2}$. Find the minimum value of $|AB|$.

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