
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 parametric equation of line $l$ is $\begin{cases} x=2+t \\ y=-1+ \sqrt{3}t \end{cases}$ (where $t$ is the parameter), and the polar equation of curve $C$ is $\rho=2\sin \theta+4\cos \theta$.  
$(1)$ Find the rectangular coordinate equation and the parametric equation of curve $C$;  
$(2)$ Find the length of the chord cut by line $l$ on curve $C$.

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