
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, and an eccentricity of $e$. The line $l: y=ex+a$ intersects the $x$-axis and $y$-axis at points $A$ and $B$, respectively. $M$ is a common point between line $l$ and ellipse $C$, and $\overrightarrow{AM}= \frac{3}{4} \overrightarrow{AB}$.
(1) Calculate the eccentricity $e$ of the ellipse.
(2) If line $l$ is moved one unit to the right to obtain $l'$, and the length of the chord intercepted by $l'$ on ellipse $C$ is $\frac{5}{4}$, find the equation of ellipse $C$.

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