(4) Let $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ be 7 distinct prime numbers, and two of $a, b, c$ sum to 800. Let $d$ be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of $d$.
Let's assume $a0
$$

Therefore,
$$
c<a+b<a+c<b+c
$$

Since $a+b, a+c, b+c$ include one number that is 800, we have
$$
c<800
$$

Given that 799 is $17 \times 47$ and 798 is not a prime number, while 797 is a prime number, it follows that $c \leqslant 797, d \geqslant 1594$.
On the other hand, when $a+b=800$, note that
$$
\begin{array}{c}
a=5, b=795 \\
a=7, b=793=13 \times 61 \\
a=11, b=789=3 \times 263
\end{array}
$$

are not all prime numbers, and thus do not meet the requirements of the problem.
However, when $a=13, b=787$, both are prime numbers. In this case, $a+b-c=3, a+c-b=23$ are also prime numbers. It is easy to verify that $b+c-a=1571$ and $a+b+c=1597$ are also prime numbers.
In summary, the maximum possible value of $d$ is 1594.