Consider a circle \(S_{1}\),

\[x^{2}+y^{2}+1-2x(\cos\theta - \sin\theta)-2y(\cos\theta + \sin\theta)=0\]

Now, consider another circle \(S_{2}\), centred at \(P=(-\cos\theta - \sin\theta, \cos\theta - \sin\theta) \) such that \(S_{1}\) internally touches \(S_{2}\).

Draw a pair of tangents \(T_{1} \) and \(T_{2}\) from \(P\) to \(S_{1}\). Let these tangents meet the circle \(S_{2}\) at points \(A\) and \(B\) as shown. From a point \(R\) on \(S_{2}\), draw chords \(RA\) and \(RB\) with lengths \(l_{1} , l_{2}\), respectively to \(S_{2}\).

Then, there exists a certain \(\alpha\) such that one of

\[\large \left| \cos^{-1}\frac{l_{1}}{6} + \cos^{-1}\frac{l_{2}}{6}\right| \quad \text{ or } \quad \left|\cos^{-1}\frac{l_{1}}{6}-\cos^{-1}\frac{l_{2}}{6} \right|\]

is equal to \(\alpha\) regardless of the value of \(R\). Find the value of this constant \(\alpha\).

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