
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
In the plane, $A(3,0)$, $B(-1,0)$, $C$ is a moving point. If $\overrightarrow{AC} \cdot \overrightarrow{BC} = 5$.  
$(1)$ Find the equation of the trajectory of point $C$.  
$(2)$ If the line $l: x-y+3=0$ intersects the curve $C$ at $M$ and $N$, find the length of $|MN|$.

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