
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 with center C(1, 2) that passes through the point (0, 1),
(Ⅰ) Write the standard equation of circle C;
(Ⅱ) Draw a tangent line to circle C from point P(2, -1) and find the equation of the tangent line as well as the length of the line segment from P to the point of tangency on the circle.

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