Not an usual Trigonometric Summation!

Calculus Level 5

\[\large{L = \lim_{n \to \infty} \sum_{k=1}^n \dfrac{1}{2^k} \tan\left(\dfrac{\pi}{3\cdot2^{k+1}}\right)}\]

If \(L\) can be represented as:

\[\large{\dfrac{A}{\pi^B} - \sqrt{C}}\]

where \(A,B,C\) are positive integers, find the value of \(A+B+C\)?


Here is a similar problem : Limited Trigonometry.
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