
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 the equation of circle M is $x^2+(y-2)^2=1$, and the equation of the line \( l \) is $x-2y=0$. Point \( P \) lies on the line \( l \), and lines \( PA \) and \( PB \) are tangents to circle M at points A and B, respectively.
1. If \( \angle APB = 60^\circ \), determine the coordinates of point P.
2. If the coordinates of point P are \( (2, 1) \), and line through P intersects the circle M at points C and D, find the equation of line CD when \( CD = \sqrt{2} \).
3. Determine whether the circle passing through points A, P, and M passes through a fixed point other than M. If so, find the coordinates of this point; if not, provide a reason.

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