
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 $C: x^{2}+y^{2}+Dx+Ey+F=0$ whose center is in the second quadrant and has a radius of $\sqrt{2}$, and the circle $C$ is tangent to the line $3x+4y=0$ and the y-axis.

$(1)$ Find $D$, $E$, and $F$;

$(2)$ If the line $x-y+2\sqrt{2}=0$ intersects circle $C$ at points $A$ and $B$, find $|AB|$.

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