Lucky Number

A positive integer is called "lucky number" if it has exactly $2014$ positive divisors and can divided by $2014$ . Let $k$ denote the number of all possibilities of "lucky number" and $a_{1},a_{2},...,a_{k}$ are all possibilities of "lucky number" . If there is a positive integer $p$ which satisfies $2014^p|a_{1}a_{2}a_{3}...a_{k}$ ,then determine the maximum value of $p$ which possible.