Always bringing me back

Consider the simple linear map $$\mathcal{L} : x \rightarrow ax+b$$ where $$a=4/7$$, and $$b = 5$$. Starting at some initial point $$X_0$$, we move to our next point $$X_1$$ according to $$X_1 \rightarrow \mathcal{L}(X_0)$$.

Suppose we iterate the map infinitely many times, ending up at $$X_\infty$$, what is the value of $$X_\infty$$?

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