Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b)\equiv 8\pmod{p} \] for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.

[i]Proposed by Justin Stevens and Evan Chen[/i]
Given the problem, we need to find the maximum possible value of \( k \) when \( m^p \) is divided by \( p^4 \). Let's break down the solution step-by-step.

1. **Initial Condition Analysis:**
   The given condition is:
   \[
   a^n(b+c) + b^n(a+c) + c^n(a+b) \equiv 8 \pmod{p}
   \]
   for each nonnegative integer \( n \).

2. **Setting \( n = 0 \):**
   When \( n = 0 \), the equation simplifies to:
   \[
   (b+c) + (a+c) + (a+b) = 2(a+b+c) \equiv 8 \pmod{p}
   \]
   This implies:
   \[
   a + b + c \equiv 4 \pmod{p}
   \]

3. **Defining \( t_n \):**
   Let \( t_n = a^n + b^n + c^n \). The given condition can be rewritten as:
   \[
   t_{n+1} = a^{n+1} + b^{n+1} + c^{n+1} = 4t_n - 8
   \]
   This implies a recurrence relation:
   \[
   t_{n+1} = 5t_n - 4t_{n-1}
   \]

4. **Solving the Recurrence Relation:**
   The characteristic equation for the recurrence relation \( t_{n+1} = 5t_n - 4t_{n-1} \) is:
   \[
   x^2 - 5x + 4 = 0
   \]
   Solving this, we get the roots:
   \[
   x = 1 \quad \text{and} \quad x = 4
   \]
   Therefore, the general solution for \( t_n \) is:
   \[
   t_n = \alpha \cdot 1^n + \beta \cdot 4^n = \alpha + \beta \cdot 4^n
   \]

5. **Determining \( \alpha \) and \( \beta \):**
   From the initial conditions:
   \[
   t_0 = a^0 + b^0 + c^0 = 3
   \]
   \[
   t_1 = a + b + c = 4
   \]
   Substituting these into the general solution:
   \[
   t_0 = \alpha + \beta = 3
   \]
   \[
   t_1 = \alpha + 4\beta = 4
   \]
   Solving these equations, we get:
   \[
   \alpha + \beta = 3
   \]
   \[
   \alpha + 4\beta = 4
   \]
   Subtracting the first equation from the second:
   \[
   3\beta = 1 \implies \beta = \frac{1}{3}
   \]
   \[
   \alpha = 3 - \frac{1}{3} = \frac{8}{3}
   \]

6. **Possible Values of \( a, b, c \):**
   Since \( a, b, c \) are positive integers and \( \{a, b, c\} \subseteq \{1, 4\} \), we must have:
   \[
   a = 1, b = 4, c = 4 \quad \text{or permutations thereof}
   \]

7. **Finding \( m \):**
   Given \( a = 1, b = 4, c = 4 \):
   \[
   a^p + b^p + c^p \equiv 1^p + 4^p + 4^p \equiv 1 + 4 + 4 = 9 \pmod{p}
   \]
   Since \( p = 5 \):
   \[
   m \equiv 9 \pmod{5} \implies m = 4
   \]

8. **Finding \( k \):**
   We need to find \( m^p \mod p^4 \):
   \[
   m^p = 4^5 = 1024
   \]
   \[
   1024 \mod 625 = 399
   \]

Thus, the maximum possible value of \( k \) is \( 399 \).

The final answer is \( \boxed{399} \)