
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 that the center of circle $C$ is at the origin of the coordinate system, and it is tangent to the line $l_{1}$: $x-y-2 \sqrt {2}=0$.
(1) Find the equation of circle $C$;
(2) Find the length of the chord $AB$ intercepted by circle $C$ on the line $l_{2}$: $4x-3y+5=0$;
(3) Draw two lines tangent to circle $C$ passing through point $G(1,3)$. Let $M$ and $N$ be the points of tangency. Find the equation of the line $MN$.

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