Question 1135367
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For solving such problems, there is beautiful Internet site

    https://www.dcode.fr  

which has a section devoted to permutations  https://www.dcode.fr/permutations-generator.


It asked me what items I want to permute, and I pointed 5 letters A, G, A, I and N.


Then it asked me, if I want to have only <U>DISTINGUISHABLE</U> words as the output, and I answered "Yes".


In response, it generated 60 = {{{5!/2}}}  words placing them in column.


This output can be ordered in ascending or descending order - I selected "descending", exactly as in a dictionary 
(it is also called lexicographic order).


From the list, I copied the 50-th line.


It is the word  "NAAIG" ,  which is your    <U>ANSWER</U>.
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Probably, I could solve the problem without the help from this solver,
but in this case I decided to save my brain and my time.