3 (Full score: 50 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x-y+z-w=2, \\
x^{2}-y^{2}+z^{2}-w^{2}=6, \\
x^{3}-y^{3}+z^{3}-w^{3}=20, \\
x^{4}-y^{4}+z^{4}-w^{4}=66 .
\end{array}\right.
$$
Let $p=x+z, q=xz$, we have
$$
\begin{array}{l}
p^{2}=x^{2}+z^{2}+2 q, \\
p^{3}=x^{3}+z^{3}+3 p q, \\
p^{4}=x^{4}+z^{4}+4 p^{2} q-2 q^{2},
\end{array}
$$

Similarly, let $s=y+w, t=yw$, we have
$$
\begin{array}{l}
s^{2}=y^{2}+w^{2}+2 t, \\
s^{3}=y^{3}+w^{3}+3 s t \\
s^{4}=y^{4}+w^{4}+4 s^{2} t-2 t^{2}
\end{array}
$$
(10 points)
In this notation system, the first equation of the original system is
$$
p=s+2 \text{. }
$$

Thus,
$$
\begin{array}{l}
p^{2}=s^{2}+4 s+4, \\
p^{3}=s^{3}+6 s^{2}+12 s+8, \\
p^{4}=s^{4}+8 s^{3}+24 s+32 s+16 .
\end{array}
$$

Now substituting the expressions for $p^{2}, p^{3}, p^{4}$ and $s^{2}, s^{3}, s^{4}$, we get
$$
\begin{array}{l} 
x^{2}+z^{2}+2 q=y^{2}+w^{2}+2 t+4 s+4, \\
x^{3}+z^{3}+3 p q=y^{3}+w^{3}+3 s t+6 s^{2}+12 s+8, \\
x^{4}+z^{4}+4 p^{2} q-2 q^{2}= y^{4}+w^{4}+4 s^{2} t-2 t^{2} \\
+8 s^{3}+24 s+32 s+16,
\end{array}
$$

Using the second to fourth equations of the original system to simplify, we get
$$
\begin{array}{c}
q=t+2 s-1, \\
p q=s t+2 s^{2}+4 s-4 \\
2 p^{2} q-q^{2}=2 s^{2} t-t^{2}+4 s^{3}+12 s^{2}+16 s-25
\end{array}
$$

Substituting (3.1) and (3.2) into (3.3), we get
$$
t=\frac{s}{2}-1 .
$$

Substituting (3.5) into (3.2), we get
$$
q=\frac{5}{2} s-2 .
$$

Substituting (3.1), (3.5), and (3.6) into (3.4), we get $s=2$. Therefore, $t=0, p=4$, and $q=3$.

Thus, $x, z$ and $y, w$ are the roots of the equations $X^{2}-4 X+3=0$ and $Y^{2}-2 Y=0$, respectively, i.e.,
(30 points)
$$
\left\{\begin{array}{l}
x=3, \\
z=1
\end{array} \text { or } \left\{\begin{array}{l}
x=1, \\
z=3
\end{array}\right.\right.
$$

and
$$
\left\{\begin{array}{l}
y=2, \\
w=0
\end{array} \text { or } \left\{\begin{array}{l}
y=0, \\
w=2
\end{array}\right.\right.
$$

In detail, the system of equations has the following four solutions: $x=3, y=2, z=1, w=0$; or $x=3, y=0, z=1, w=2$; or $x=1, y=2, z=3, w=0$; or $x=1, y=0, z=3, w=2$.
(50 points)