
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $k$ .

*Proposed by Justin Stevens and Evan Chen*

Present the answer in LaTex format: \boxed{Your answer}
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