
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 $x^2+y^2-4x+2y-3=0$ and a point $M(4, -8)$ outside the circle.  
(1) A secant line passing through $M$ intersects the circle at points $A$ and $B$. If $|AB|=4$, find the equation of line $AB$.  
(2) A tangent line passing through $M$ touches the circle at points $C$ and $D$. Find the length of the tangent and the equation of the line containing $C$ and $D$.

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