
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
In a coordinate system, the pole of the polar coordinate system coincides with the origin of the rectangular coordinate system, and the polar axis coincides with the non-negative semi-axis of the x-axis. The line C₁ is given by the parametric equations $\begin{cases} x=1+t\cos\alpha \\ y=1+t\sin\alpha \end{cases}$ (where t is a parameter), and the curve C₂ is given by the polar equation ρ² - 2ρcosθ - 8 = 0.
(I) Find the rectangular coordinate equation of the curve C₂.
(II) The line C₁ intersects the curve C₂ at two points A and B. Find the ordinary equation of the trajectory of the midpoint M of AB.

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