[THOUGHT] We are given: a + b + c = 151 a^2 + b^2 + c^2 = 10939 a^3 + b^3 + c^3 = 957871

We need a^4 + b^4 + c^4.

Let S1 = a+b+c = 151 S2 = a^2+b^2+c^2 = 10939 S3 = a^3+b^3+c^3 = 957871

We can use Newton's identities to relate power sums to elementary symmetric polynomials.

Let e1 = a+b+c = S1 = 151 e2 = ab + bc + ca e3 = abc

Newton's identities: S1 = e1 S2 = e1S1 - 2e2 => 10939 = 151151 - 2e2 => 10939 = 22801 - 2e2 => 2e2 = 22801 - 10939 = 11862 => e2 = 5931

S3 = e1S2 - e2S1 + 3e3 => 957871 = 15110939 - 5931151 + 3e3 Compute stepwise: 15110939 = 15110939 = let's compute. Better to compute in Python.

Then we can find e3.

Then S4 = a^4+b^4+c^4 can be computed using recurrence: S4 = e1S3 - e2S2 + e3S1 - 4e4? Wait careful: Newton's identities for n=3: For k>=4, we have S_k = e1S_{k-1} - e2S_{k-2} + e3S_{k-3} - ... but since we have only three variables, for k=4: S4 = e1S3 - e2S2 + e3S1 - 4e4? Actually e4 = 0 because there are only three variables (product of four distinct variables is zero). So indeed: S4 = e1S3 - e2S2 + e3S1

Thus we need e3.

Let's compute with Python. [/THOUGHT] [PYTHON] import sympy

S1 = 151 S2 = 10939 S3 = 957871

Compute e2 from S2 = e1S1 - 2e2

e1 = S1 e2 = (e1*S1 - S2) / 2 print("e2 =", e2)

Compute e3 from S3 = e1S2 - e2S1 + 3*e3

e3 = (S3 - e1S2 + e2S1) / 3 print("e3 =", e3)

Compute S4 = e1S3 - e2S2 + e3*S1

S4 = e1S3 - e2S2 + e3*S1 print("S4 = a^4+b^4+c^4 =", S4) [/PYTHON]

[THOUGHT] From the Python output: e2 = 5931.0 e3 = 67221.0 S4 = a^4 + b^4 + c^4 = 89909683.0

The result is an integer: 89909683. Thus the final answer is 89909683. [/THOUGHT] \boxed{89909683}