
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
The complex number $z$ satisfies $|2z-1+i|=4$, and $w=z(1-i)+2+i$.  
(1) Find the trajectory $C$ of point $P$ corresponding to $w$ in the complex plane.  
(2) In the complex plane, a tangent line is drawn from point $Q(0,4)$ to the trajectory $C$, touching at points $A$ and $B$. Find the equation of line $AB$.

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