# 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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