Let $\phi(t)$ satisfy the linear singular integral equation

$$\left(t^{2}+t-1\right) \phi(t)-\frac{t^{2}-t-1}{\pi i} \oint_{L} \frac{\phi(\tau) d \tau}{\tau-t}-\frac{1}{\pi i} \int_{L}\left(\tau+\frac{1}{\tau}\right) \phi(\tau) d \tau=t-1, \quad t \in L,$$

where $\oint$ denotes the principal value integral and $L$ denotes a counterclockwise smooth closed contour, enclosing the origin but not the points $\pm 1$.

(a) Formulate the associated Riemann-Hilbert problem.

(b) For this Riemann-Hilbert problem, find the index, the homogeneous canonical solution and the solvability condition.

(c) Find $\phi(t)$.