Fibonacci Dials

• There are six dials, each with four values.  Each value is a unique positive integer (for a total of 24 unique values.)
• The dials are arranged left to right.  The first two dials spin freely.  The other dials may only turn to a new value if the two previous dials sum to that value. 
• There are no unused values.  There exists some way for each dial to turn to all four of its values, and the values on the first two dials can all be used in a sum.  (The first dial's values can all be used with values from the second dial to obtain a sum on the third dial.  The values on the second dial can be used to obtain a sum on either the third or fourth dial.  For example, the first three dials could be: {1,2,3,4}; {5,6,7,8}; {9,10,11,12}.  The {1,2,3,4} can be summed with {8,7,6,5} to obtain "9" on the third dial, and the {10,11,12} can all be obtained through other sums.)
• The largest value (of all 24) is as small as possible.

Source: Original.