
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 equations of curve $C$ as $\begin{cases} & x=3+\sqrt{5}\cos \alpha \\ & y=1+\sqrt{5}\sin \alpha \end{cases}$ (where $\alpha$ is the parameter), and establishing a polar coordinate system with the origin of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis.

$(1)$ Find the polar equation of curve $C$;

$(2)$ If the polar equation of a line is $\rho \sin \theta -\rho \cos \theta =1$, find the length of the chord cut by the line on curve $C$.

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