
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 curve $C_{1}$ is: $\left\{\begin{array}{c}x=t+\frac{1}{t}\\ y=2(t-\frac{1}{t})\end{array}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is: $ρ-\frac{1}{\sinθ-3\cosθ}=0$. <br/> $(1)$ Find the rectangular coordinate equation of curve $C_{2}$; <br/> $(2)$ Given that points $P$ and $Q$ are moving points on curves $C_{1}$ and $C_{2}$ respectively, find the minimum value of $|PQ|$ and the coordinates of point $P$ when the minimum value is reached.

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