
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 $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{1}{2}$, the ellipse $E$ intersects with the parabola $y^{2}= \frac{9}{4}x$ at points $M$ and $N$. The line $MN$ passes through the right focus $F_{2}$ of the ellipse $E$.
(1) Find the standard equation of the ellipse $E$;
(2) Given that the left focus of the ellipse $E$ is $F_{1}$, and the left and right vertices are $A$ and $B$, respectively. A line $l$ passing through point $F_{1}$ intersects with the ellipse $E$ at points $C$ and $D$. Denote the areas of $\triangle ABD$ and $\triangle ABC$ as $S_{1}$ and $S_{2}$, respectively. Find the maximum value of $|S_{1}-S_{2}|$.

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