Let $k$ be the smallest positive integer with the following property:
There are distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial
$$
p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)
$$
has exactly $k$ nonzero coefficients.
Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.

Let's think step by step and output the final answer within \boxed{}.
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