To solve this problem, we will need to:

1. Find the lengths $| \vec{a} |$, $| \vec{b} |$, and $| \vec{c} |$.
2. Calculate the areas of the parallelograms formed by pairs of vectors $\vec{a}$ and $\vec{b}$, $\vec{b}$ and $\vec{c}$, and $\vec{a}$ and $\vec{c}$.
3. Compute the volume of the parallelepiped formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.

We are given: $\vec{a} = (x, 2, 2), \quad \vec{b} = (4, -1, 5), \quad \vec{c} = (3, y, -2),$ $(\vec{a}, \vec{b}) = 20, \quad (\vec{b}, \vec{c}) = -3.$

Step 1: Find the lengths of $\vec{a}$, $\vec{b}$, and $\vec{c}$

The length (magnitude) of a vector $\vec{v} = (v_1, v_2, v_3)$ is given by: $|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}.$

1. Finding $| \vec{a} |$: $|\vec{a}| = \sqrt{x^2 + 2^2 + 2^2} = \sqrt{x^2 + 4 + 4} = \sqrt{x^2 + 8}.$

2. Finding $| \vec{b} |$: $|\vec{b}| = \sqrt{4^2 + (-1)^2 + 5^2} = \sqrt{16 + 1 + 25} = \sqrt{42}.$

3. Finding $| \vec{c} |$: $|\vec{c}| = \sqrt{3^2 + y^2 + (-2)^2} = \sqrt{9 + y^2 + 4} = \sqrt{y^2 + 13}.$

Step 2: Calculate the dot products to find $x$ and $y$

The dot product of two vectors $\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$.

1. Dot product $\vec{a} \cdot \vec{b} = 20$: $\vec{a} \cdot \vec{b} = x \cdot 4 + 2 \cdot (-1) + 2 \cdot 5 = 4x - 2 + 10 = 4x + 8.$ Setting this equal to 20: $4x + 8 = 20 \Rightarrow 4x = 12 \Rightarrow x = 3.$

2. Dot product $\vec{b} \cdot \vec{c} = -3$: $\vec{b} \cdot \vec{c} = 4 \cdot 3 + (-1) \cdot y + 5 \cdot (-2) = 12 - y - 10 = 2 - y.$ Setting this equal to -3: $2 - y = -3 \Rightarrow y = 5.$

With $x = 3$ and $y = 5$, we can now find the magnitudes:

1. Final magnitude of $\vec{a}$: $|\vec{a}| = \sqrt{3^2 + 2^2 + 2^2} = \sqrt{9 + 4 + 4} = \sqrt{17}.$

2. Magnitude of $\vec{b}$: $|\vec{b}| = \sqrt{42}.$

3. Final magnitude of $\vec{c}$: $|\vec{c}| = \sqrt{3^2 + 5^2 + (-2)^2} = \sqrt{9 + 25 + 4} = \sqrt{38}.$

Step 3: Calculate areas of the parallelograms

The area of a parallelogram formed by two vectors $\vec{u}$ and $\vec{v}$ is given by $|\vec{u} \times \vec{v}|$.

1. Area of parallelogram formed by $\vec{a}$ and $\vec{b}$: [ \vec{a} \times \vec{b} = \begin{vmatrix} \hat