Let $a, b$ be two real numbers such that $a b

Question

Let $a, b$ be two real numbers such that $a b<0$. If the complex number $\frac{1+a i}{b+i}$ is of unit modulus and $a+i b$ lies on the circle $|z-1|=|2 z|$, then a possible value of $\frac{1+[a]}{4 b}$, where $[t]$ is greatest integer function, is: [JEE Main 2023, 1 Feb (Shift 2)]

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