[THOUGHT] We need to find S5 = a^5 + b^5 + c^5 + d^5 given symmetric sums S1 = a+b+c+d = 151, S2 = a^2+b^2+c^2+d^2 = 7989, S3 = a^3+b^3+c^3+d^3 = 511201, S4 = a^4+b^4+c^4+d^4 = 35858673.

Let p = a+b+c+d = S1 = 151. Let q = ab+ac+ad+bc+bd+cd. Let r = abc+abd+acd+bcd. Let s = abcd.

We have Newton's identities relating power sums P_k = a^k+b^k+c^k+d^k to elementary symmetric sums e1 = p, e2 = q, e3 = r, e4 = s, e5 = 0 (since there are only 4 variables).

Newton identities for n=4: P1 = e1 = p. P2 = e1P1 - 2 e2 = pP1 - 2q. But P1 = p, so P2 = p^2 - 2q. Given P2 = 7989, p=151. Thus 7989 = 151^2 - 2q → 151^2 = 22801 → 7989 = 22801 - 2q → 2q = 22801 - 7989 = 14812 → q = 7406.

P3 = e1P2 - e2P1 + 3 e3 = pP2 - qP1 + 3r. Given P3 = 511201, p=151, P2=7989, q=7406, P1=151. Compute: pP2 = 1517989 = ? 1518000 = 1,208,000, minus 15111 = 1661, so 1,208,000 - 1661 = 1,206,339. Check: Actually 1517989 = 151(8000-11) = 1,208,000 - 1661 = 1,206,339. Now qP1 = 7406151 = 7406150 + 7406 = 1,110,900 + 7406 = 1,118,306. Then pP2 - q*P1 = 1,206,339 - 1,118,306 = 88,033. Then P3 = 88,033 + 3r = 511201 → 3r = 511201 - 88033 = 423168 → r = 423168 / 3 = 141056.

P4 = e1P3 - e2P2 + e3P1 - 4 e4 = pP3 - qP2 + rP1 - 4s. Given P4 = 35858673. Compute pP3 = 151511201 = ? 151511201 = 151(511200 + 1) = 151511200 + 151 = 1515112100 = (1515112)100. Better: 151511201 = 511201150 + 511201 = 76,680,150 + 511,201 = 77,191,351. Check: Actually 511201150 = 76,680,150, plus 511,201 = 77,191,351. Now qP2 = 74067989 = ? 74068000 = 59,248,000 minus 740611 = 81,466 → 59,248,000 - 81,466 = 59,166,534. Then rP1 = 141056151 = 141056150 + 141056 = 21,158,400 + 141,056 = 21,299,456. Now compute pP3 - qP2 + rP1 = 77,191,351 - 59,166,534 + 21,299,456. First: 77,191,351 - 59,166,534 = 18,024,817. Then add 21,299,456 → 18,024,817 + 21,299,456 = 39,324,273. Then P4 = 39,324,273 - 4s = 35,858,673. Thus 39,324,273 - 4s = 35,858,673 → 4s = 39,324,273 - 35,858,673 = 3,465,600 → s = 866,400.

Now we have e1=151, e2=7406, e3=141056, e4=866400.

Now we want P5 = a^5+b^5+c^5+d^5. Newton identity for P5: P5 = e1P4 - e2P3 + e3P2 - e4P1 + 5 e5, but e5=0. Thus P5 = pP4 - qP3 + rP2 - sP1. Plug numbers: pP4 = 151 * 35,858,673. qP3 = 7406 * 511,201. rP2 = 141056 * 7989. sP1 = 866400 * 151.

We'll compute stepwise in Python to avoid mistakes. [/THOUGHT] [PYTHON] p = 151 q = 7406 r = 141056 s = 866400 P1 = 151 P2 = 7989 P3 = 511201 P4 = 35858673

Compute P5 = pP4 - qP3 + rP2 - sP1

term1 = p * P4 term2 = q * P3 term3 = r * P2 term4 = s * P1

P5 = term1 - term2 + term3 - term4 print("term1 =", term1) print("term2 =", term2) print("term3 =", term3) print("term4 =", term4) print("P5 =", P5) [/PYTHON]

[THOUGHT] The Python computation gives P5 = 2624775001. This is the value of a^5 + b^5 + c^5 + d^5. [/THOUGHT] \boxed{2624775001}