
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 intersecting the $y$-axis at $(0, 2\sqrt{3})$, and a parabola $C_2$ with its vertex at the origin and focus coinciding with the left focus of ellipse $C_1$.

1. Find the standard equations of ellipse $C_1$ and parabola $C_2$.
2. Two perpendicular lines passing through point $(1, 0)$ intersect 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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