
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, 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 chord AB that is intercepted on circle C by the line $l_2: 4x - 3y + 5 = 0$;
(3) Through point G(1, 3), draw two lines tangent to circle C, 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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