[THOUGHT] Let me break down this problem step by step:

1. First, I need to find the values of a, b, c, d that satisfy the given system of equations.
2. Then construct F(x) = (x-a)(x-b)(x-c)(x-d).
3. Find all quadruples (u,v,w,t) with 0 ≤ u,v,w,t ≤ 8 satisfying the three conditions.
4. Sum up u² + v² + w² + t² for all such quadruples.

Let me start by finding a, b, c, d using Newton's identities to relate power sums to elementary symmetric polynomials. [/THOUGHT] [PYTHON] from itertools import combinations_with_replacement, product from math import gcd from functools import reduce

First, find a, b, c, d

We have power sums: p1 = 35, p2 = 439, p3 = 6377, p4 = 98803

Using Newton's identities to find elementary symmetric polynomials e1, e2, e3, e4

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

Newton's identities:

p1 = e1

p2 = e1p1 - 2e2

p3 = e1p2 - e2p1 + 3*e3

p4 = e1p3 - e2p2 + e3p1 - 4e4

e1 = p1 # = 35

p2 = e1p1 - 2e2

439 = 3535 - 2e2

439 = 1225 - 2*e2

2*e2 = 1225 - 439 = 786

e2 = (e1 * p1 - p2) // 2 # = 393

p3 = e1p2 - e2p1 + 3*e3

6377 = 35439 - 39335 + 3*e3

6377 = 15365 - 13755 + 3*e3

6377 = 1610 + 3*e3

3*e3 = 4767

e3 = (p3 - e1p2 + e2p1) // 3

p4 = e1p3 - e2p2 + e3p1 - 4e4

98803 = 356377 - 393439 + 158935 - 4e4

e4 = (e1p3 - e2p2 + e3*p1 - p4) // 4

print(f"e1 = {e1}, e2 = {e2}, e3 = {e3}, e4 = {e4}")

So F(x) = x^4 - e1x^3 + e2x^2 - e3*x + e4

F(x) = x^4 - 35x^3 + 393x^2 - 1589*x + e4

Let's verify by finding the actual roots

We need to find positive integers a, b, c, d with sum 35

Let's search for them

found = [] for a in range(1, 33): for b in range(a, 34): for c in range(b, 35): d = 35 - a - b - c if d >= c and d >= 1: if a2 + b2 + c2 + d2 == 439: if a3 + b3 + c3 + d3 == 6377: if a4 + b4 + c4 + d4 == 98803: found.append((a, b, c, d)) print(f"Found: a={a}, b={b}, c={c}, d={d}")

print(f"Solutions found: {found}") [/PYTHON]

[THOUGHT] Now I have a, b, c, d = 2, 5, 11, 17 and F(x) = x⁴ - 35x³ + 393x² - 1589x + 1870.

Let me continue to find all quadruples (u,v,w,t) satisfying the three conditions and compute the sum. [/THOUGHT] [PYTHON] from itertools import product from math import gcd from functools import reduce

F(x) = x^4 - 35x^3 + 393x^2 - 1589*x + 1870

def F(x): return x4 - 35*x3 + 393x**2 - 1589x + 1870

Precompute F values for 0 to 8

F_values = {x: F(x) for x in range(9)} print("F values:", F_values)

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

def gcd_of_tuple(t): return reduce(gcd, t)

U = [] for u in range(9): for v in range(9): for w in range(9): for t in range(9): # Condition 1: u+v+w+t is even if (u + v + w + t) % 2 != 0: continue # Condition 2: F(u)+F(v)+F(w)+F(t) divisible by 12 if (F_values[u] + F_values[v] + F_values[w] + F_values[t]) % 12 != 0: continue # Condition 3: gcd(u,v,w,t) = 1 if gcd_of_tuple((u, v, w, t)) != 1: continue U.append((u, v, w, t))

print(f"Number of valid quadruples: {len(U)}")

Compute S

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

[THOUGHT] The Python code found all valid quadruples satisfying the three conditions and computed the sum S = 56040. [/THOUGHT]

\boxed{56040}