[THOUGHT] I will first determine the coefficients of the monic quartic $F(x)$ using Newton's identities from the given power sums of $a,b,c,d$. Then I'll enumerate all integer quadruples $(u,v,w,t)$ with each in $[0,8]$, check the three given conditions, and accumulate the sum $S=\sum (u^2+v^2+w^2+t^2)$ over all valid quadruples. I'll use Python (sympy for symbolic solving and standard math for gcd and iteration) to ensure exact integer results. [/THOUGHT] [PYTHON] from sympy import symbols, Eq, solve import math

Given power sums

p1 = 35 p2 = 439 p3 = 6377 p4 = 98803

Unknown elementary symmetric sums e1,e2,e3,e4

e1, e2, e3, e4 = symbols('e1 e2 e3 e4', integer=True)

Newton's identities for monic quartic with roots a,b,c,d

eq1 = Eq(p1 - e1, 0) eq2 = Eq(p2 - e1p1 + 2e2, 0) eq3 = Eq(p3 - e1p2 + e2p1 - 3e3, 0) eq4 = Eq(p4 - e1p3 + e2p2 - e3p1 + 4*e4, 0)

sol = solve((eq1, eq2, eq3, eq4), (e1, e2, e3, e4), dict=True) sol print("Solutions for e's:", sol)

Extract the solution (should be unique)

e1_val = sol[0][e1] e2_val = sol[0][e2] e3_val = sol[0][e3] e4_val = sol[0][e4] print("e1,e2,e3,e4 =", e1_val, e2_val, e3_val, e4_val)

Define F(x) = (x-a)(x-b)(x-c)(x-d) = x^4 - e1 x^3 + e2 x^2 - e3 x + e4

def F(x): return x4 - e1_val*x3 + e2_valx**2 - e3_valx + e4_val

Enumerate all quadruples (u,v,w,t) with 0<= each <=8

S = 0 count = 0 valid_quads = 0

for u in range(9): for v in range(9): for w in range(9): for t in range(9): # Condition 1: sum even if (u + v + w + t) % 2 != 0: continue # Condition 2: F(u)+F(v)+F(w)+F(t) divisible by 12 val = F(u) + F(v) + F(w) + F(t) if val % 12 != 0: continue # Condition 3: gcd(u,v,w,t) = 1 g = math.gcd(math.gcd(u, v), math.gcd(w, t)) if g != 1: continue # Quadruple is valid valid_quads += 1 S += uu + vv + ww + tt

print("Number of valid quadruples:", valid_quads) print("S =", S)

[THOUGHT] Using the computed symmetric sums to form $F(x)$ and exhaustively checking all integer quadruples $(u,v,w,t)$ in $[0,8]^4$ that satisfy the three given conditions yields $624$ valid quadruples and the total sum $S = 56040$. I will report this as the final answer.