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

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

(2) If a line (n) with a slope of $k(k\ne 0)$ intersects the ellipse (C) at points A and B, and ${{k}_{OA}}$, $k$, and ${{k}_{OB}}$ form an arithmetic sequence, and there is a point $M\left(1,1\right) $, find the maximum value of ${S}_{∆ABM} $.

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