Let $y = f(x)$ be a thrice differentiable function on $(-5, 5)$. Let the tangents to the curve $y = f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27\int_1^3 \left(f'(t)\right)^2 + 1\right) f''(t)\, dt = \alpha + \beta\sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha + \beta$ equals\\
(1) $-14$\\
(2) 26\\
(3) $-16$\\
(4) 36