
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 circle C passes through points M(0, -2) and N(3, 1), and that the center of circle C lies on the line x + 2y + 1 = 0.
(Ⅰ) Find the equation of circle C;
(Ⅱ) Construct the tangent to circle C that passes through point (6, 3) and find the equation of the tangent line;
(Ⅲ) Let line l be represented by y = x + m, and suppose that line l intersects circle C at chord AB. If circle C<sub>1</sub> with diameter AB passes through the origin, find the equation of line l.

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