Video solution:

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Written solution:

After inserting $N$ into the list, there will be $6$ total numbers. That is even, so the median will be the average of the middle two numbers.

All of the answer choices are at least $7,$ so when they are inserted into the list, the middle two numbers will be $7$ and $7.$ It is convenient that the median will always be $7,$ no matter which answer choice is picked.

The mean becomes twice the median, which is $2 \times 7 = 14.$

To have a total of $6$ numbers with mean $14,$ their sum must become $6 \times 14 = 84.$

The sum of the original $5$ numbers is $$2 + 6 + 7 + 7 + 28 = 50,$$ so the answer is $84 - 50 = 34,$ which is choice E.