
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 an ellipse $C$ with foci $F_{1}(-2\sqrt{2}, 0)$, $F_{2}(2\sqrt{2}, 0)$, and passing through point $A(3, 0)$.  
(Ⅰ) Find the standard equation of ellipse $C$.   
(Ⅱ) Let the line $y=x+2$ intersect ellipse $C$ at two points $M$ and $N$. Find the coordinates of the midpoint $P$ of segment $MN$.

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