
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 M whose center is on the line $x-2y+4=0$, and it intersects the x-axis at points A(-5, 0) and B(1, 0).
(Ⅰ) Find the equation of circle M;
(Ⅱ) Find the equation of the tangent line to circle M passing through point C(1, 2);
(Ⅲ) Given D(-3, 4), and point P moves on circle M, find the equation of the trajectory of another vertex Q of the parallelogram with AD and AP as a pair of adjacent sides.

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