
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 $C$ are 
$$
\begin{cases} 
x=3+ \sqrt {5}\cos \alpha \\
y=1+ \sqrt {5}\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}),
$$
with the origin of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system.  
$(1)$ Find the polar equation of curve $C$;  
$(2)$ If the polar equation of a line is $\sin \theta-\cos \theta= \frac {1}{\rho }$, find the length of the chord cut from curve $C$ by the line.

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