
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 center at the origin $O$ and foci on the x-axis, it passes through the point $(1, \frac{\sqrt{2}}{2})$. Suppose that the two foci of the ellipse $C$ and its two vertices form a square.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Suppose that line $l$ (with a defined slope) intersects the ellipse $C$ at points $A$ and $B$, and the area of $\triangle AOB$ is $\frac{\sqrt{2}}{2}$. Determine whether the product of the slopes of $OA$ and $OB$ is a constant when the slopes of the lines exist; if so, find the constant value, if not, explain the reason.

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