Brilli the bug

Calculus Level 3

Brilli the Bug has set out on a journey of infinite steps starting at the origin of the xyxy-plane. It moves in the following manner:

  1. After each, nnth step, it turns 9090^\circ counter-clockwise
  2. Each nnth step is of length DnD_{n} where Dn D_{n} is given by Dn=2(n+1)2D_{n} = \dfrac{2}{(n+1)^{2}} for n0 n \geq 0 .

If the final displacement of brilli from the starting is given by 1αβGγ+πη\dfrac{1}{\alpha}\sqrt{ \beta G^{\gamma} + \pi^{\eta}} , find α+β+γ+η \alpha +\beta +\gamma +\eta .

Notation: GG denotes the Catalan's constant.

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