
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 $(xOy)$, the parabola $(C)$ is defined by the equation $x^2=2py (p > 0)$ with focus $F$. Point $A$ lies on $(C)$ such that $|AO|=|AF|=\frac{3}{2}$.
(I) Find the equation of $(C)$;
(II) Let line $(l)$ intersect $(C)$ at points $P$ and $Q$. If the $y$-coordinate of the midpoint of line segment $PQ$ is $1$, find the maximum area of $\triangle OPQ$.

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