
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 $C_1: x^2 + y^2 - 2x - 4y + 4 = 0$ and a line $l: x + 2y - 4 = 0$ that intersect at points A and B:
(Ⅰ) Find the length of the chord AB;
(Ⅱ) If circle $C_2$ passes through points E(1, -3) and F(0, 4) and the common chord of circles $C_1$ and $C_2$ is parallel to the line $2x + y + 1 = 0$, find the equation of circle $C_2$.

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