Long live, bacteria!

A certain species of bacteria either die, split into two or three bacteria. All splits are exact copies. The probabilities of dying, splitting into two and splitting into three are \(\frac 14\), \(\frac 12\) and \(\frac 14\) respectively.

If the probability that it survives for infinite time is \(\dfrac {a-\sqrt b}c \), where \(a,b,c\) are coprime positive integers, find \(a + b +c\).

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