[THOUGHT] I need to find natural numbers $a, b, c, d$ satisfying the given system of equations, and then compute $a^5 + b^5 + c^5 + d^5$.

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

• $p_1 = a + b + c + d = 151$
• $p_2 = a^2 + b^2 + c^2 + d^2 = 7989$
• $p_3 = a^3 + b^3 + c^3 + d^3 = 511201$
• $p_4 = a^4 + b^4 + c^4 + d^4 = 35858673$

And let $e_1, e_2, e_3, e_4$ be the elementary symmetric polynomials.

Newton's identities give us:

• $p_1 = e_1$
• $p_2 = e_1 p_1 - 2e_2$
• $p_3 = e_1 p_2 - e_2 p_1 + 3e_3$
• $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1 - 4e_4$
• $p_5 = e_1 p_4 - e_2 p_3 + e_3 p_2 - e_4 p_1$

Let me compute the elementary symmetric polynomials first, then find $p_5$. [/THOUGHT] [PYTHON]

Given power sums

p1 = 151 p2 = 7989 p3 = 511201 p4 = 35858673

Newton's identities to find elementary symmetric polynomials

p1 = e1

e1 = p1

p2 = e1p1 - 2e2 => e2 = (e1*p1 - p2) / 2

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

p3 = e1p2 - e2p1 + 3e3 => e3 = (p3 - e1p2 + e2*p1) / 3

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

p4 = e1p3 - e2p2 + e3p1 - 4e4 => e4 = (e1p3 - e2p2 + e3*p1 - p4) / 4

e4 = (e1 * p3 - e2 * p2 + e3 * p1 - p4) // 4

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

Now compute p5 using Newton's identity

p5 = e1p4 - e2p3 + e3p2 - e4p1

p5 = e1 * p4 - e2 * p3 + e3 * p2 - e4 * p1

print(f"p5 = a^5 + b^5 + c^5 + d^5 = {p5}") [/PYTHON]

[THOUGHT] Using Newton's identities, I computed the elementary symmetric polynomials from the given power sums, and then calculated $p_5 = a^5 + b^5 + c^5 + d^5 = 2624775001$. [/THOUGHT]

\boxed{2624775001}