
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 line $l$: $y=x$ and the circle $C$: $(x-2)^{2}+(y-4)^{2}=10$.

$(1)$ Find the coordinates of the intersection points between line $l$ and circle $C$.

$(2)$ Find the length of the chord that circle $C$ cuts off from line $l$.

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