
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 its center at the origin and its focus $F$ on the $x$-axis, the eccentricity $e= \frac { \sqrt {3}}{2}$, and a point $Q( \sqrt {2}, \frac { \sqrt {2}}{2})$ on the ellipse $C$.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ If a line $n$ with slope $k(k\neq 0)$ intersects the ellipse $C$ at points $A$ and $B$, and $k_{OA}$, $k$, $k_{OB}$ form an arithmetic sequence, with point $M(1,1)$, find the maximum value of $S_{\triangle ABM}$.

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