
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
In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ has its center at the origin $(O)$, its foci on the $x$-axis, its minor axis length is $2$, and its eccentricity is $\frac{\sqrt{2}}{2}$.

(1) Find the equation of ellipse $(C)$.

(2) Let $A$ and $B$ be two arbitrary points on the ellipse $(C)$ such that the area of $\triangle AOB$ is $\frac{\sqrt{6}}{4}$. Let $E$ be the midpoint of segment $AB$. The ray $OE$ intersects the ellipse $(C)$ at point $O$. Assume $\overrightarrow{OP} = t \overrightarrow{OE}$, find the value of the real number $t$.

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