
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 $\begin{cases} x=3+ \sqrt{10}\cos \alpha \\ y=1+ \sqrt{10}\sin \alpha \end{cases}$ (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 $C$ and explain what trajectory it represents;

$(2)$ 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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