
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 $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $e= \frac{1}{2}$, and its positive semi-axis intersects with the $y$-axis at $(0,2 \sqrt {3})$. A parabola $C\_2$ has its vertex at the origin and its focus coincides with the left focus of the ellipse $C\_1$.

1. Find the standard equations of the ellipse $C\_1$ and the parabola $C\_2$.
2. Two perpendicular lines passing through $(1,0)$ intersect with parabola $C\_2$ at four points, forming a quadrilateral. Find the minimum area of this quadrilateral.

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