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The following picture shows a portion of the New Year's Day decorations on the street, as the communities welcome \( 2014 \). The decorations comprise of an equilateral triangle, colored yellow, and a square, colored green, which share vertex B. When measured, the distance from vertex \( E \) to the ground \( (EH) \) is \( 8 \) miles, while the distance from vertex \( C \) to the ground \( (CG) \) is \( 2+\sqrt{3} \) miles. Additionally, the flags are joined such that \( \angle FBC=120^\circ \).

Let the sum of the three areas (not decorated by flags) \( ACG, BAD, \) and \( HED \) be \( k \) \( \text{mi}^2 \). Find \( \lceil k \rceil\).

Note:\( \lceil x \rceil \) denotes the least integer greater than or equal to \( x \).

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