I am carrot +_+

We have a magical carrot multiplier machine, which multiplies an input carrot into at most 6 carrots, and minimum just 1 (gives the carrot back). But it can multiply only original carrots, the ones which were not put into the machine ever.

There are \(286\) carrots of \(143\) different shades such that there are exactly \(2\) carrots of each shade. All of these carrots after treating with the machine, are filled in a bag.

There are \(11^n\) different compositions possible for this bag. Find \(n\).

Details and Assumptions:-

\(\bullet\) The carrots formed by multiplying one particular carrot are identical (due to magic of the multiplier).

\(\bullet\) The \(143\) different shades are well distinguishable, no 2 look alike.

This problem is a part of the set Vegetable Combinatorics


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