
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 $F1(-2\sqrt{2},0)$ and $F2(2\sqrt{2},0)$, and the major axis is $6$.

$(1)$ Find the standard equation of the ellipse $C$;  

$(2)$ Suppose the line $y=x+2$ intersects the ellipse $C$ at points $A$ and $B$, find the coordinates of the midpoint of segment $AB$.

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