
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
A circle C with equation $(x-1)^2+y^2=1$ passes through point P(3,1). Draw two tangents PA and PB to circle C, with A and B as the points of tangency.  
(1) Find the equation of line AB;  
(2) If point Q is any point on circle C, find the equation of the trajectory of the midpoint M of PQ.

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