Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that  
  
$P(1) = P(3) = P(5) = P(7) = a,$  and  
  
$P(2) = P(4) = P(6) = P(8) = -a.$ 
  
What is the smallest possible value of $a$?

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