Let $P(x)$ be a polynomial with integer coefficients satisfying $$(x^2+1)P(x-1)=(x^2-10x+26)P(x)$$ for all real numbers $x.$ Find the sum of all possible values of $P(0)$ between $1$ and $5000,$ inclusive.

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