Differential Equations: JEE Main 2025 Mathematics question

Let $g$ be a differentiable function such that $\displaystyle\int_0^x g(t)\,dt = x - \int_x^\pi g(t)\,dt$, $x\ge0$, and let $y=y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x\,g(x)$ for $x\in[0,\tfrac{\pi}{2}]$. If $y(0)=0$, then $y\bigl(\tfrac{\pi}{3}\bigr)$ is equal to: