
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
An ellipse $C_{1}$ has the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, its major axis is equal to the diameter of a circle $C_{2}$ with equation $x^{2}+y^{2}=4$, and the eccentricity of $C_{1}$ is $\frac{1}{2}$. Two perpendicular lines, $l_{1}$ and $l_{2}$, pass through the point $M(1,0)$. $l_{1}$ intersects $C_{1}$ at points $A$ and $B$, while $l_{2}$ intersects $C_{2}$ at points $C$ and $D$.

(1) Find the standard equation of $C_{1}$.

(2) When the area of quadrilateral $ACBD$ is $\frac{12}{7} \sqrt{14}$, find the slope $k (k > 0)$ of line $l_{1}$.

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