Graze me, Hit me

There are two parallel planes, each inclined to the horizontal at an angle \(\theta\). A particle is projected from a point midway between the foot of the two planes so that it grazes one of the planes and strikes the other at a right angle. Find the angle of projection of the projectile.

If this angle can be expressed as \( \tan^{-1} (a \cot \theta) + \theta\), submit your answer as \( a^2 + 2\sqrt2\).

Note: Diagram is not necessarily up to scale.

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