
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
Given the curve $C_{1}$ in the Cartesian coordinate system with the parametric equations $\begin{cases} x=\dfrac{\sqrt{5}}{5}t \\ y=\dfrac{2\sqrt{5}}{5}t-1 \end{cases}$ (where $t$ is the parameter), and establishing a polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, there is a curve $C_{2}$: $\rho=2\cos \theta-4\sin \theta$  
$(1)$ Convert the equation of $C_{1}$ into a standard equation (general form), and find the standard Cartesian coordinate equation of $C_{2}$.  
$(2)$ Calculate the distance between the two intersection points of curves $C_{1}$ and $C_{2}$.

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