[THOUGHT] I need to find natural numbers $a, b, c$ satisfying the given system of equations, and then compute $a^4 + b^4 + c^4$.

Let me use Newton's identities to relate power sums to elementary symmetric polynomials.

Let:

• $p_1 = a + b + c = 151$
• $p_2 = a^2 + b^2 + c^2 = 10939$
• $p_3 = a^3 + b^3 + c^3 = 957871$

And elementary symmetric polynomials:

• $e_1 = a + b + c$
• $e_2 = ab + bc + ca$
• $e_3 = abc$

Newton's identities give us:

• $p_1 = e_1$
• $p_2 = e_1 p_1 - 2e_2$
• $p_3 = e_1 p_2 - e_2 p_1 + 3e_3$
• $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$

From these, I can find $e_1, e_2, e_3$ and then compute $p_4 = a^4 + b^4 + c^4$.

Let me calculate this step by step. [/THOUGHT] [PYTHON]

Given values

p1 = 151 p2 = 10939 p3 = 957871

Newton's identities

p1 = e1

e1 = p1

p2 = e1 * p1 - 2 * e2

e2 = (e1 * p1 - p2) / 2

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

p3 = e1 * p2 - e2 * p1 + 3 * e3

e3 = (p3 - e1 * p2 + e2 * p1) / 3

e3 = (p3 - e1 * p2 + e2 * p1) // 3

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

Now compute p4 using Newton's identity

p4 = e1 * p3 - e2 * p2 + e3 * p1

p4 = e1 * p3 - e2 * p2 + e3 * p1

print(f"p4 = a^4 + b^4 + c^4 = {p4}")

Let's verify by finding a, b, c

They are roots of: t^3 - e1t^2 + e2t - e3 = 0

import numpy as np

coeffs = [1, -e1, e2, -e3] roots = np.roots(coeffs) print(f"Roots: {roots}")

Verify the sums

a, b, c = sorted([int(round(r.real)) for r in roots]) print(f"a, b, c = {a}, {b}, {c}") print(f"Verification:") print(f"a + b + c = {a + b + c}") print(f"a^2 + b^2 + c^2 = {a2 + b2 + c2}") print(f"a^3 + b^3 + c^3 = {a3 + b3 + c3}") print(f"a^4 + b^4 + c^4 = {a4 + b4 + c**4}") [/PYTHON]

[THOUGHT] I found the natural numbers $a, b, c$ that satisfy the given equations: $a = 21$, $b = 33$, $c = 97$.

The verification confirms:

• $a + b + c = 21 + 33 + 97 = 151$ ✓
• $a^2 + b^2 + c^2 = 441 + 1089 + 9409 = 10939$ ✓
• $a^3 + b^3 + c^3 = 9261 + 35937 + 912673 = 957871$ ✓

And the answer we need: $a^4 + b^4 + c^4 = 21^4 + 33^4 + 97^4 = 194481 + 1185921 + 88529281 = 89909683$ [/THOUGHT]

\boxed{89909683}