Abnormally Normal

Calculus Level 3

Consider the graph plotted by the equation

kx+yk+xx=xk+ky+yyk^x+y^k+x^x=x^k+k^y+y^y

where x>0x>0, y>0y>0, and kR+k \in \mathbb{R^+} is a constant.

Let Gk,tG_{k,t} be the gradient of the normal to a curve of the graph at the point (t,t)(t,t) where tZ+t \in \mathbb{Z^+}, tkt \geq k is a real positive integer. Determine the value of

k=22016(t=kk+2015Gk,t)\prod_{k=2}^{2016} \left( \prod_{t=k}^{k+2015} G_{k,t} \right)

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