Graphs of the Other Trigonometric Functions · Challenge problem

For the graph of $y=a\sec x+d$ with $a>0$, the lowest point of each upward branch is $10$ units above the highest point of each downward branch, and the graph passes through $\left(\dfrac{\pi}{3},7\right)$. Find the exact value of $y$ at $x=\dfrac{\pi}{4}$.