
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 circle $C$ at two distinct points $P$ and $Q$, and the circle with diameter $PQ$ passes through the origin, find the y-intercept of the line;
(3) Draw two tangents to 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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