
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 $C$ with its center at the origin $O$, and it is tangent to the line $l_{1}: x-y-2\sqrt{2}=0$.
$(1)$ Find the length of the chord $AB$ intercepted by the circle $C$ on the line $l_{2}: 4x-3y+5=0$;
$(2)$ If a line perpendicular to $l_{1}$ intersects the circle $C$ at two distinct points $P$ and $Q$, and a circle with $PQ$ as its diameter passes through the origin, find the y-intercept of the line;
$(3)$ Draw two tangents to the circle $C$ from point $G(1,3)$, with the points of tangency being $M$ and $N$, respectively. Find the equation of line $MN$.

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