To encode a message, James first replaces each letter within the message with its corresponding number, where A = $1$, B = $2$, $\dots$ , Y = $25$, and Z = $26$. Next, James multiplies each number by $3$ and then subtracts $5$, and continues this process a total of $n$ times. For example, when $n = 2$ the letter D is encoded to the number $16$. If James encoded a four-letter message to the four numbers $367$, $205$, $853$, $1339$, what is the value of $n$ that he used?