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Let $$a_n:=\dfrac23(4^n + 2).$$ Note that $\sum_{k=1}^{n-1}4^k=4^n-a_n$ and hence $P(|S_{n-1}|\le4^n-a_n)=1$. So, the equality $X_n=S_n-S_{n-1}$ and the triangle inequality imply $$P(|S_n|\ge a_n)\ge P(|X_n|=4^n,|S_{n-1}|\le4^n-a_n)=1$$ and hence $$P(|S_n|\ge a_n)=1.$$ On the other hand, $$P(|S_n|=a_n)\ge P(X_n=4^n,S_{n-1}=-(4^n-a_n)) \\ =P(X_n=4^n,X_{n-1}=-4^{n-1},\dots,X_1=-4^1)=2^{-n}>0$$ and hence for all $r>a_n$ we have $$P(|S_n|\ge r)\le P(|S_n|>a_n)\le1-2^{-n}<1.$$ So, $$A_n=a_n.\quad\Box$$