Let $𝑋_1,𝑋_2, … , 𝑋_𝑛$ be a sequence of independent random variables with $𝑃(𝑋_𝑛 = 4^𝑛) = 𝑃(𝑋_𝑛 = −4^𝑛) = \frac12$. Let $𝑆_𝑛 = 𝑋_1 + 𝑋_2 + ⋯ + 𝑋_𝑛$. If $A_n=\sup\, \{𝑟 ∈ \Bbb R: 𝑃(|𝑆_𝑛| ≥ 𝑟) = 1 \}$ for $𝑛 ∈ \Bbb N$, show that $A_𝑛 = \dfrac23(4^𝑛 + 2)$?

for both extreme cases,

$S_𝑛=(4/3)(4^𝑛-1)$ and $S_𝑛=(-4/3)(4^𝑛-1)$

How to proceed with the proof after this?