
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 $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=\frac{2+t}{6},\\ y=\sqrt{t}\end{array}\right.$ ($t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\left\{{\begin{array}{l}{x=-\frac{{2+s}}{6},}\\{y=-\sqrt{s}}\end{array}}\right.$ ($s$ is the parameter).<br/>$(1)$ Write down the Cartesian equation of $C_{1}$;<br/>$(2)$ Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis. The polar coordinate equation of curve $C_{3}$ is $2\cos \theta -\sin \theta =0$. Find the Cartesian coordinates of the intersection points of $C_{3}$ with $C_{1}$ and $C_{3}$ with $C_{2}$.

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