
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 $x^2+y^2=4$ with a fixed point $A(2,0)$, and $B(1,1)$ is a point inside the circle. $P$ and $Q$ are moving points on the circle.  
(1) Find the equation of the trajectory of the midpoint of segment $AP$.  
(2) If $\angle PBQ=90^\circ$, find the equation of the trajectory of the midpoint of segment $PQ$.

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