
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with its left and right foci being $F_1$ and $F_2$ respectively, and point $M(0, 2)$ is a vertex of the ellipse. $\triangle F_1MF_2$ is an isosceles right triangle.  
(1) Find the equation of ellipse $C$;  
(2) Let point $P$ be a moving point on ellipse $C$, find the trajectory equation of the midpoint $Q$ of segment $PM$;  
(3) Draw lines $MA$ and $MB$ from point $M$ to intersect the ellipse at points $A$ and $B$ respectively, with the slopes of the two lines being $k_1$ and $k_2$, and $k_1 + k_2 = 8$. Investigate whether line $AB$ passes through a fixed point, and explain the reason.

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