Three-cushion billiards

On a frictionless plane, ball AA moves with a velocity of vv to collide elastically with two stopped balls BB and CC sequentially, as shown above. The masses of the three balls are the same as m.m. As indicated by the red arrows, the direction of ball AA's movement changes by an angle of 4545° each time there is a collision. What is the sum of the quantities of the impulses acting on ball AA by balls BB and C?C?

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