
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 $\frac{1}{2}$, ellipse $E$ intersects with the parabola $y^2 = \frac{9}{4}x$ at points $M$ and $N$, and the line $MN$ exactly 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 the ellipse $E$ at points $C$ and $D$. Let the areas of $\triangle ABD$ and $\triangle ABC$ be $S_1$ and $S_2$ respectively, find the maximum value of $|S_1 - S_2|$.

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