Let $u, f$, and g be functions defined for all real numbers x, such that

$$
\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).
$$

If $u(x)=g(x+a)-f(x+b)+f(x+c)-f(x+d)+f(x+e)$, what is the value of $a+b+c+d+e$.

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