A natural number is called a *prime power* if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. 

Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

Let's think step by step and output the final answer within \boxed{}.
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