
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 a circle $C: x^{2}+y^{2}-2y-4=0$ and a line $l: mx-y+1-m=0$ where $m\in \mathbb{R}$.  
$(1)$ Write down the coordinates of the center and the radius of the circle $C$, and determine the relationship between the line $l$ and the circle $C$.  
$(2)$ Let the line $l$ intersect the circle $C$ at points $A$ and $B$. If the slope of the line $l$ is $120^{\circ}$, find the length of chord $AB$.

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