Infinitely nested inverse

Calculus Level 3

Let $$f : [-1, 1] \to [0, \pi]$$ be defined as $$f(x) = \arccos(x)$$. Define $$f_{n}(x)$$ as $$\underbrace{f \circ f \circ f \circ \cdots \circ f}_{\text{n times}}$$ if it's defined.

For some $$x \in [-1, 1]$$, $$\displaystyle \lim_{n \to \infty}{f_{n}(x)} = A$$, a real number. Evaluate $$A$$ to $$3$$ decimal places.

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