
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ ($a>b>0$) with its center at the origin $O$ and left and right foci $F_1$ and $F_2$, respectively. Let $P$ be a point on the ellipse such that $PF_2$ is perpendicular to the x-axis, $|PF_{2}|= \frac{1}{2}$, and the eccentricity is $\frac{\sqrt{3}}{2}$.
(I) Find the standard equation of the ellipse;
(II) A line $l$ passing through the left focus with a slope of 45° intersects the ellipse $C$ at points $A$ and $B$. Calculate the area of triangle $AOB$.

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