Graphs of the Other Trigonometric Functions · Superchallenge problem

Build the function $y = a\sec(bx - c) + d$ with $a > 0$, $b > 0$ and $0 \le c < 2\pi$ that has period $4\pi$, adjacent vertical asymptotes at $x = \dfrac{4\pi}{3}$ and $x = \dfrac{10\pi}{3}$, whose upward U-shapes have minimum value $4$ and whose downward U-shapes have maximum value $2$, and whose branch between those two asymptotes is a downward U. Find $a$, $b$, $c$ and $d$.