# Planet Brilliantia

On planet Brilliantia, there are two types of creatures: mathematicians and non-mathematicians.

Mathematicians tell the truth $$\frac{6}{7}$$ of the time and lie only $$\frac{1}{7}$$ of the time, while non-mathematicians tell the truth $$\frac{1}{5}$$ of the time and lie $$\frac{4}{5}$$ of the time.

It is also known that there is a $$\frac{2}{3}$$ chance a creature from Brilliantia is a mathematician and a $$\frac{1}{3}$$ chance that it is a non-mathematician, but there is no way of differentiating from these two types.

You are visiting Brilliantia on a research trip. During your stay, you come across a creature who states that it has found a one line proof for Fermat's Last Theorem. Immediately after that, a second creature shows up and states that the first creature's statement was a true one.

If the probability that the first creature's statement was actually true is $$\frac{a}{b}$$, for some coprime positive integers $$a, b$$, find the value of $$b - a$$.

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