Consider five positive integers a, b, c, d, and p, with gcd(a,b,c)=1, that satisfy this system of equations:

• a, b, and c are different positive integers.
• a²+b²+c²=d²
• a+b+c=p²

Determine the four smallest values of a+b+c+d+p.

*** Computer program/excel solver assisted solutions are welcome, but a semi-analytic (p&p and hand calculator) methodology is preferred.