OSN 2021 Problem 3

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 $3 \le i \le n$ .