Fourier base - not as you know it!

Number Theory Level pending

A natural number \(n\) is called foury if there exsits a natural number \(b\) such that in base \(b\) all the digits of \(n\) are fours, e.g. \(4\) is foury and so is \(624\) as \(624_{10}=4444_{5}\). Clearly, if \(n\) is foury then \(n\equiv0\pmod 4\). There exists a smallest foury number \(N\) such that for all \(n\geq N\) if \(n\equiv0\pmod 4\) then \(n\) is foury.

A natural number \(n\) is called unfoury if in any natural base \(b\), none of the digits of \(n\) is four. E.g. \(1\) is unfoury. There exists a largest unfoury number \(M\).

Find \(N\) and \(M\). Your answer will be of the form \(N.M\) (with a decimal point between them). E.g. if you think that \(N=4\) and \(M=1\) then you should enter \(4.1\).

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