For how many pairs \((a,b)\) from \((1,2)\), \((3,5)\), \((5,7)\), \((7,11)\), does the polynomial \(P(x)=x^5+ax^4+bx^3+bx^2+ax+1\) have exactly one real root?

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