Question 131
We write numbers $1, 2, \dots, 2014$ in some order and denote them with $a_1, a_2, \dots, a_{2014}$. We then rewrite them in some new order and denote them with $b_1, b_2, \dots, b_{2014}$. Find the largest positive integer $k$ such that, regardless of the chosen orders, the number $3^k$ will certainly divide the number
$$
(a_1^2 - b_1^2)(a_2^2 - b_2^2)\dots(a_{2014}^2 - b_{2014}^2).
$$
$$
(a_1^2 - b_1^2)(a_2^2 - b_2^2)\dots(a_{2014}^2 - b_{2014}^2).
$$