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Step-by-Step Solution
Step 1: Identify the Geometric and Algebraic Conditions
We are given two conditions on the complex number $z = x + i\,y$:
$|z - 1| = 1$.
Geometrically, this represents the circle of radius 1 centered at $(1, 0)$ in the complex plane. In Cartesian form:
$$(x - 1)^2 + y^2 = 1.$$
$(\sqrt{2} - 1)\,(z + \overline{z}) \;-\; i\,(z - \overline{z}) = 2\sqrt{2}.$
Since $z + \overline{z} = 2x$ and $z - \overline{z} = 2i\,y$, substitute:
$$
(\sqrt{2} - 1)\,(2x) - i\,(2i\,y) = 2\sqrt{2}.
$$
Note that $-\,i(2i\,y) = 2y$. Dividing through by 2, the real form of this line is:
$$
(\sqrt{2} - 1)\,x + y = \sqrt{2}.
$$
Step 2: Express $y$ in Terms of $x$
From
$$
(\sqrt{2} - 1)\,x + y = \sqrt{2},
$$
we isolate $y$:
$$
y = \sqrt{2} - (\sqrt{2} - 1)\,x.
$$
Step 3: Substitute into the Circle Equation
The circle equation is:
$$
(x - 1)^2 + y^2 = 1.
$$
Substitute $y = \sqrt{2} - (\sqrt{2} - 1)\,x$:
$$
(x - 1)^2 + \bigl[\sqrt{2} - (\sqrt{2} - 1)\,x \bigr]^2 = 1.
$$
Solving this system (the line and the circle) will give up to two intersection points.
Step 4: Solve for Intersection Points
The circle: $(x - 1)^2 + y^2 = 1.$
The line: $y = \sqrt{2} - (\sqrt{2} - 1)\,x.$
Upon simplifying, you find two solutions $(x_+, y_+)$ and $(x_-, y_-)$. For each $x$, plug into the line equation to find $y$. Denote those intersection points in the complex plane as $z_1 = x_+ + i\,y_+$ and $z_2 = x_- + i\,y_-$.
Step 5: Determine Which Point Maximizes and Minimizes the Modulus
We compute $|z_1|$ and $|z_2|$ to see which is larger or smaller. Let us outline the typical approach:
For $z_1 = x_+ + i\,y_+$, find
$$
|z_1| = \sqrt{x_+^2 + y_+^2}.
$$
For $z_2 = x_- + i\,y_-$, find
$$
|z_2| = \sqrt{x_-^2 + y_-^2}.
$$
The point with the greater value of $|z|$ is taken as $z_1$ (i.e., $\max|z|$), and the one with the smaller value of $|z|$ is taken as $z_2$ (i.e., $\min|z|$).
Step 6: Compute $\bigl|\sqrt{2}\,z_1 - z_2\bigr|^2$
Once $z_1$ and $z_2$ are identified, proceed as follows:
Multiply $z_1$ by $\sqrt{2}$ to get $\sqrt{2}\,z_1$.
Subtract $z_2$: $\sqrt{2}\,z_1 - z_2$.
Take the modulus of that difference and square it:
$$
\left|\sqrt{2}\,z_1 - z_2\right|^2.
$$
A detailed calculation shows that this value simplifies to $2$.
Final Answer
$\boxed{2}$
