
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 that the center of ellipse $C$ is at the origin, its focus $F$ is on the $x$-axis, and its eccentricity $e=\frac{\sqrt{3}}{2}$. Also, point $Q(\sqrt{2} , \frac{\sqrt{2}}{2})$ is on ellipse $C$.

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

(2) If a line $n$ with slope $k (k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, and the slopes of $OA$, $AB$, and $OB$ form an arithmetic sequence, find the area of triangle $ABM$ in terms of $k$, where $M(1,1)$ is a given point;

(3) Find the maximum area of triangle $ABM$ from part (2).

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