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We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the firstn digits of its continued fraction expansion performs in the order ofn^4 mathematical operations. The construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main difficulty is to control the length of these subintervals. To achieve this we adapt and combine known metric theorems on continued fractions and on expansions in integers bases.