
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 that the equation of circle $M$ is $(x^2+(y-2)^2=1)$, the equation of line $l$ is $(x-2y=0)$, and point $P$ is on line $l$. A tangent line $PA$ and $PB$ is drawn through point $P$, with tangent points $A$ and $B$.

(1) If the coordinates of point $P$ are $(0,0)$, find $\angle APB$;

(2) If the coordinates of point $P$ are $(2,1)$, draw a line through $P$ intersecting circle $M$ at points $C$ and $D$. When $CD=\sqrt{2}$, find the equation of line $CD$;

(3) Does the circle passing through points $A$, $P$, and $M$ also pass through a fixed point other than point $M$? If so, find the coordinates of this fixed point; if not, explain the reason.

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