
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
On the circle $O: x^2 + y^2 = 4$, take any point $P$. Draw a perpendicular line segment $PQ$ from point $P$ to the y-axis, with $Q$ being the foot of the perpendicular. When $P$ moves on the circle, the trajectory of the midpoint $G$ of line segment $PQ$ is $C$.
(Ⅰ) Find the equation of $C$;
(Ⅱ) Line $l$ intersects circle $O$ at points $M$ and $N$, and intersects curve $C$ at points $E$ and $F$. If $|MN| = \frac {8 \sqrt {5}}{5}$, determine whether $\angle EOF$ is a constant value. If yes, find this constant value; if no, explain why.

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