The magnetic field \( B \) at a point midway between two parallel currents in opposite directions is:
\[ B = \frac{\mu_0 I}{2 \pi a} \times 2 = \frac{4 \pi \times 10^{-7} \times 10}{\pi \times (2.5 \times 10^{-2})} \] \[ = 16 \times 10^{-5} = 160 \, \mu T \]
Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is: