Find out maximum as well as minimum

Algebra Level 5

\[ \large{ p\leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4)\leq q }\]

Let the real numbers \(a,b,c,d\) satisfy the relations \(a+b+c+d=6\) and \(a^2+b^2+c^2+d^2=12\). If above inequality is satisfied for some positive integers \(p\) and \(q\). Find the value of \(p+q\)

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