[THOUGHT] We need to solve step by step.

First, we have positive integers a,b,c,d satisfying:

1. a+b+c+d = 35
2. a^2+b^2+c^2+d^2 = 439
3. a^3+b^3+c^3+d^3 = 6377
4. a^4+b^4+c^4+d^4 = 98803

We can find a,b,c,d by solving system of equations. Since they are positive integers, we can try to find them via polynomial roots: they are roots of quartic F(x) = (x-a)(x-b)(x-c)(x-d) = x^4 - e1 x^3 + e2 x^2 - e3 x + e4, where e1 = sum a = 35, 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 can compute e2, e3, e4 using Newton's identities.

Let s1 = sum a_i = 35 s2 = sum a_i^2 = 439 s3 = sum a_i^3 = 6377 s4 = sum a_i^4 = 98803

Newton's identities: p1 = s1 = e1 p2 = s1 e1 - 2 e2 = s2 => 35*35 - 2 e2 = 439 => 1225 - 2 e2 = 439 => 2 e2 = 1225 - 439 = 786 => e2 = 393

p3 = s1 e2 - s2 e1 + 3 e3 = s3 => 35393 - 43935 + 3 e3 = 6377 => 35*(393 - 439) + 3 e3 = 6377 => 35*(-46) + 3 e3 = 6377 => -1610 + 3 e3 = 6377 => 3 e3 = 6377 + 1610 = 7987 => e3 = 7987/3 = 2662.333... That's not integer? Wait, check: 7987/3 = 2662.333..., but e3 should be integer because a,b,c,d are integers. Did I make a mistake? Let's recompute.

Newton's identities: For monic polynomial x^n - e1 x^{n-1} + e2 x^{n-2} - ... + (-1)^n en = 0 with roots r_i. Define s_k = sum r_i^k. Then: s1 - e1 = 0 => e1 = s1. s2 - e1 s1 + 2 e2 = 0 => e2 = (s1^2 - s2)/2. s3 - e1 s2 + e2 s1 - 3 e3 = 0 => e3 = (s3 - e1 s2 + e2 s1)/3. s4 - e1 s3 + e2 s2 - e3 s1 + 4 e4 = 0 => e4 = (-s4 + e1 s3 - e2 s2 + e3 s1)/4.

Let's compute stepwise.

Given s1=35, s2=439, s3=6377, s4=98803.

e1 = s1 = 35. e2 = (s1^2 - s2)/2 = (1225 - 439)/2 = 786/2 = 393. Now e3 = (s3 - e1 s2 + e2 s1)/3 = (6377 - 35439 + 39335)/3. Compute 35439 = 35400=14000, 3539=1365, total 15365? Wait 35439 = 35*(440-1)=15400-35=15365. 39335 = 39330=11790, 3935=1965, total 13755. So numerator = 6377 - 15365 + 13755 = 6377 - 15365 = -8988, plus 13755 = 4767. 4767/3 = 1589 exactly? 31589=4767 yes. So e3 = 1589.

Now e4 = (-s4 + e1 s3 - e2 s2 + e3 s1)/4. Compute: s4=98803, e1 s3 = 356377 = 356000=210000, 35377=13195, total 223195. e2 s2 = 393439 = 393400=157200, 39339=15327, total 172527. e3 s1 = 158935 = 158930=47670, 1589*5=7945, total 55615. So numerator = -98803 + 223195 - 172527 + 55615 = compute stepwise: -98803 + 223195 = 124392. 124392 - 172527 = -48135. -48135 + 55615 = 7480. 7480/4 = 1870. Thus e4 = 1870.

So polynomial F(x) = x^4 - 35 x^3 + 393 x^2 - 1589 x + 1870.

Now we need to find integer roots a,b,c,d positive integers. Factor polynomial? We can find roots via integer root theorem: factors of 1870: ±1, ±2, ±5, ±10, ±11, ±17, ±22, ±34, ±55, ±85, ±110, ±170, ±187, ±374, ±935, ±1870.

Try positive integers: Let's test with Python.

Then we have F(x) defined.

Next, we need to consider integer quadruples (u,v,w,t) with 0 ≤ u,v,w,t ≤ 8 (so each from 0 to 8 inclusive). Conditions:

1. u+v+w+t is even.
2. F(u)+F(v)+F(w)+F(t) divisible by 12.
3. gcd(u,v,w,t)=1.

Let U be set of such quadruples.

Compute S = sum_{(u,v,w,t) in U} (u^2+v^2+w^2+t^2).

We need to enumerate all quadruples (u,v,w,t) with each 0..8 (9^4 = 6561 possibilities), filter by conditions, sum squares.

We'll write Python code to compute.

First, find a,b,c,d to verify F(x). Then compute F(x) for x=0..8.

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

Given sums

s1 = 35 s2 = 439 s3 = 6377 s4 = 98803

Compute elementary symmetric polynomials

e1 = s1 e2 = (s1**2 - s2)//2 e3 = (s3 - e1s2 + e2s1)//3 e4 = (-s4 + e1s3 - e2s2 + e3*s1)//4

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

Polynomial coefficients: x^4 - e1 x^3 + e2 x^2 - e3 x + e4

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

Find integer roots

roots = [] for x in range(1, 100): val = (((coeff[0]*x + coeff[1])*x + coeff[2])*x + coeff[3])*x + coeff[4] if val == 0: roots.append(x) print("Positive integer roots:", roots)

Verify sums

a, b, c, d = roots print("a,b,c,d:", a, b, c, d) print("Check sums:") print("Sum:", a+b+c+d) print("Sum squares:", a2+b2+c2+d2) print("Sum cubes:", a3+b3+c3+d3) print("Sum fourth powers:", a4+b4+c4+d4)

Define F(x)

def F(x): return ((x - a)(x - b)(x - c)*(x - d))

Precompute F values for x=0..8

F_vals = [F(x) for x in range(9)] print("F(0..8):", F_vals)

Enumerate quadruples

U = [] sum_squares = 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 total_F = F_vals[u] + F_vals[v] + F_vals[w] + F_vals[t] if total_F % 12 != 0: continue # condition 3: gcd(u,v,w,t)=1 g = math.gcd(math.gcd(math.gcd(u, v), w), t) if g != 1: continue U.append((u, v, w, t)) sum_squares += u2 + v2 + w2 + t2

print("Number of quadruples in U:", len(U)) print("Sum of squares S:", sum_squares)

Also compute sum explicitly to verify

S_check = sum(u2+v2+w2+t2 for (u,v,w,t) in U) print("S check:", S_check) [/PYTHON]

[THOUGHT] The Python code computed the sum S = 56040. This is the final answer. [/THOUGHT] \boxed{56040}