Example 1 Given that $a, b, c, d$ are prime numbers (allowing $a, b, c, d$ to be the same), and $abcd$ is the sum of 35 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$ . [1]
Let
\[
\begin{array}{l}
a b c d=x+(x+1)+\cdots+(x+34) \\
=5 \times 7(x+17) .
\end{array}
\]

Assume \( a=5, b=7, c \leqslant d, c d=x+17 \).
Thus, \( d_{\text {min }}=5 \).
If \( d=5 \), then \( c_{\text {min }}=5 \), at this point,
\[
x=8, a+b+c+d=22 \text {; }
\]

If \( d=7 \), then \( c_{\text {min }}=3 \), at this point,
\[
x=4, a+b+c+d=22 \text {; }
\]

If \( d \geqslant 11 \), then
\[
a+b+c+d \geqslant 5+7+2+11=25 \text {. }
\]

In summary, the minimum value of \( a+b+c+d \) is 22.