[THOUGHT] We need to solve for integers a,b,c,d positive satisfying the given symmetric polynomial equations. Since they are symmetric sums, we can use Newton's identities to find elementary symmetric sums and then factor the quartic F(x). Once we have F(x), we can evaluate F(u) for u from 0 to 8 and compute sum over quadruples (u,v,w,t) with constraints.

First, find elementary symmetric sums e1 = a+b+c+d, e2 = sum_{i<j} a_i a_j, e3 = sum_{i<j<k} a_i a_j a_k, e4 = a b c d.

We know: p1 = e1 = 35 p2 = sum a_i^2 = 439 p3 = sum a_i^3 = 6377 p4 = sum a_i^4 = 98803

Newton's identities: p1 = e1 p2 = e1 p1 - 2 e2 => e2 = (e1 p1 - p2)/2 p3 = e1 p2 - e2 p1 + 3 e3 => e3 = (p3 - e1 p2 + e2 p1)/3 p4 = e1 p3 - e2 p2 + e3 p1 - 4 e4 => e4 = (e1 p3 - e2 p2 + e3 p1 - p4)/4

Let's compute with Python to avoid arithmetic errors.

Then F(x) = x^4 - e1 x^3 + e2 x^2 - e3 x + e4.

We need to find integer roots a,b,c,d? Actually a,b,c,d are positive integers, but we don't need to find them individually; we only need F(u) for u from 0 to 8. However, to compute F(u) we need e1,e2,e3,e4. But maybe we can find F(x) as a polynomial with integer coefficients. Since a,b,c,d are integers, F(x) is monic with integer coefficients.

Thus compute e1,e2,e3,e4 via Python.

Then compute F(u) for u = 0..8.

Then enumerate all quadruples (u,v,w,t) with 0 <= u,v,w,t <= 8, satisfying:

1. u+v+w+t even
2. (F(u)+F(v)+F(w)+F(t)) % 12 == 0
3. gcd(u,v,w,t) == 1 (gcd of all four numbers)

Sum of squares u^2+v^2+w^2+t^2 over those quadruples.

We must be careful: quadruple is ordered? The problem says "set of integer quadruples (u,v,w,t)" but likely ordered quadruples because condition gcd(u,v,w,t)=1 and sum of squares depends on order? Usually in such problems, quadruple is ordered unless specified otherwise. However, the sum is over all ordered quadruples satisfying conditions. Let's check: The notation (u,v,w,t) ∈ U where U is a set of quadruples. Typically in such combinatorial sums, order matters unless they say "unordered". Since they specify quadruple (u,v,w,t) with inequalities 0 ≤ u,v,w,t ≤ 8, that suggests ordered quadruples. Also gcd condition is defined on the tuple. So we treat as ordered quadruples.

But we must be careful about duplicates: if (u,v,w,t) and (v,u,w,t) are distinct unless u=v. So we enumerate all 9^4 = 6561 quadruples.

Let's implement.

We'll compute e1,e2,e3,e4, then F(x) as polynomial, evaluate at integers 0..8.

Then iterate over all quadruples, check conditions, accumulate sum of squares.

Let's code. [/THOUGHT] [PYTHON] import itertools import math

Given symmetric sums

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

Compute elementary symmetric sums e1,e2,e3,e4 using Newton's identities

e1 = p1 e2 = (e1 * p1 - p2) // 2 e3 = (p3 - e1 * p2 + e2 * p1) // 3 e4 = (e1 * p3 - e2 * p2 + e3 * p1 - p4) // 4

print("e1,e2,e3,e4:", e1, e2, e3, e4)

Coefficients of F(x) = x^4 - e1 x^3 + e2 x^2 - e3 x + e4

coeff = [1, -e1, e2, -e3, e4] print("Coefficients:", coeff)

Evaluate F(u) for u = 0..8

F_vals = [] for u in range(9): # Evaluate polynomial using Horner val = coeff[0] for c in coeff[1:]: val = val * u + c F_vals.append(val) print("F(u) for u=0..8:", F_vals)

Precompute F(u) mod 12

F_mod12 = [f % 12 for f in F_vals] print("F(u) mod 12:", F_mod12)

Precompute squares mod something? Not needed.

Precompute gcd function for four numbers

def gcd4(a,b,c,d): return math.gcd(math.gcd(a,b), math.gcd(c,d))

total_sum = 0 count = 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: sum of F mod 12 == 0 if (F_mod12[u] + F_mod12[v] + F_mod12[w] + F_mod12[t]) % 12 != 0: continue # condition 3: gcd(u,v,w,t) == 1 if gcd4(u, v, w, t) != 1: continue # All conditions satisfied total_sum += uu + vv + ww + tt count += 1

print("Count of quadruples:", count) print("Sum of squares:", total_sum) [/PYTHON]

[THOUGHT] The Python output shows that the sum of squares over all valid quadruples is 56040. Therefore, the final answer is 56040.

\boxed{56040}